Properties

Label 13.10.a.a.1.2
Level 1313
Weight 1010
Character 13.1
Self dual yes
Analytic conductor 6.6956.695
Analytic rank 11
Dimension 44
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [13,10,Mod(1,13)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(13, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("13.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: N N == 13 13
Weight: k k == 10 10
Character orbit: [χ][\chi] == 13.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 6.695465870136.69546587013
Analytic rank: 11
Dimension: 44
Coefficient field: Q[x]/(x4)\mathbb{Q}[x]/(x^{4} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x31602x2+1544x+342272 x^{4} - x^{3} - 1602x^{2} + 1544x + 342272 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2 2
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.2
Root 16.536016.5360 of defining polynomial
Character χ\chi == 13.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q24.5360q2+49.9972q3+90.0171q4+1814.98q51226.73q68707.31q7+10353.8q817183.3q944532.3q1082919.7q11+4500.61q1228561.0q13+213643.q14+90743.8q15300130.q16+374907.q17+421610.q18361501.q19+163379.q20435341.q21+2.03452e6q222.31340e6q23+517661.q24+1.34101e6q25+700774.q261.84321e6q27783807.q28649654.q292.22649e6q30+4.32521e6q31+2.06285e6q324.14575e6q339.19873e6q341.58036e7q351.54679e6q36+1.21021e7q37+8.86981e6q381.42797e6q39+1.87919e7q40+2.59960e6q41+1.06816e7q42+3.08318e6q437.46419e6q443.11872e7q45+5.67617e7q46+2.60298e7q471.50057e7q48+3.54636e7q493.29032e7q50+1.87443e7q512.57098e6q521.01982e8q53+4.52251e7q541.50497e8q559.01536e7q561.80741e7q57+1.59399e7q58+1.37562e8q59+8.16850e6q604.29579e7q611.06124e8q62+1.49620e8q63+1.03052e8q645.18375e7q65+1.01720e8q66+5.54479e7q67+3.37480e7q681.15664e8q69+3.87757e8q701.43613e8q711.77912e8q72+3.37147e7q732.96937e8q74+6.70469e7q753.25413e7q76+7.22007e8q77+3.50367e7q782.66652e8q795.44728e8q80+2.46063e8q816.37839e7q829.89542e7q833.91882e7q84+6.80447e8q857.56490e7q863.24809e7q878.58533e8q885.08803e8q89+7.65211e8q90+2.48689e8q912.08246e8q92+2.16249e8q936.38667e8q946.56116e8q95+1.03137e8q964.53920e8q978.70137e8q98+1.42483e9q99+O(q100)q-24.5360 q^{2} +49.9972 q^{3} +90.0171 q^{4} +1814.98 q^{5} -1226.73 q^{6} -8707.31 q^{7} +10353.8 q^{8} -17183.3 q^{9} -44532.3 q^{10} -82919.7 q^{11} +4500.61 q^{12} -28561.0 q^{13} +213643. q^{14} +90743.8 q^{15} -300130. q^{16} +374907. q^{17} +421610. q^{18} -361501. q^{19} +163379. q^{20} -435341. q^{21} +2.03452e6 q^{22} -2.31340e6 q^{23} +517661. q^{24} +1.34101e6 q^{25} +700774. q^{26} -1.84321e6 q^{27} -783807. q^{28} -649654. q^{29} -2.22649e6 q^{30} +4.32521e6 q^{31} +2.06285e6 q^{32} -4.14575e6 q^{33} -9.19873e6 q^{34} -1.58036e7 q^{35} -1.54679e6 q^{36} +1.21021e7 q^{37} +8.86981e6 q^{38} -1.42797e6 q^{39} +1.87919e7 q^{40} +2.59960e6 q^{41} +1.06816e7 q^{42} +3.08318e6 q^{43} -7.46419e6 q^{44} -3.11872e7 q^{45} +5.67617e7 q^{46} +2.60298e7 q^{47} -1.50057e7 q^{48} +3.54636e7 q^{49} -3.29032e7 q^{50} +1.87443e7 q^{51} -2.57098e6 q^{52} -1.01982e8 q^{53} +4.52251e7 q^{54} -1.50497e8 q^{55} -9.01536e7 q^{56} -1.80741e7 q^{57} +1.59399e7 q^{58} +1.37562e8 q^{59} +8.16850e6 q^{60} -4.29579e7 q^{61} -1.06124e8 q^{62} +1.49620e8 q^{63} +1.03052e8 q^{64} -5.18375e7 q^{65} +1.01720e8 q^{66} +5.54479e7 q^{67} +3.37480e7 q^{68} -1.15664e8 q^{69} +3.87757e8 q^{70} -1.43613e8 q^{71} -1.77912e8 q^{72} +3.37147e7 q^{73} -2.96937e8 q^{74} +6.70469e7 q^{75} -3.25413e7 q^{76} +7.22007e8 q^{77} +3.50367e7 q^{78} -2.66652e8 q^{79} -5.44728e8 q^{80} +2.46063e8 q^{81} -6.37839e7 q^{82} -9.89542e7 q^{83} -3.91882e7 q^{84} +6.80447e8 q^{85} -7.56490e7 q^{86} -3.24809e7 q^{87} -8.58533e8 q^{88} -5.08803e8 q^{89} +7.65211e8 q^{90} +2.48689e8 q^{91} -2.08246e8 q^{92} +2.16249e8 q^{93} -6.38667e8 q^{94} -6.56116e8 q^{95} +1.03137e8 q^{96} -4.53920e8 q^{97} -8.70137e8 q^{98} +1.42483e9 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q33q2163q3+1429q4+471q54529q611241q745543q829953q967831q1040140q11155479q12114244q13277653q14+83307q15+726609q16++2132181050q99+O(q100) 4 q - 33 q^{2} - 163 q^{3} + 1429 q^{4} + 471 q^{5} - 4529 q^{6} - 11241 q^{7} - 45543 q^{8} - 29953 q^{9} - 67831 q^{10} - 40140 q^{11} - 155479 q^{12} - 114244 q^{13} - 277653 q^{14} + 83307 q^{15} + 726609 q^{16}+ \cdots + 2132181050 q^{99}+O(q^{100}) Copy content Toggle raw display

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 −24.5360 −1.08435 −0.542175 0.840266i 0.682399π-0.682399\pi
−0.542175 + 0.840266i 0.682399π0.682399\pi
33 49.9972 0.356369 0.178185 0.983997i 0.442978π-0.442978\pi
0.178185 + 0.983997i 0.442978π0.442978\pi
44 90.0171 0.175815
55 1814.98 1.29869 0.649346 0.760493i 0.275043π-0.275043\pi
0.649346 + 0.760493i 0.275043π0.275043\pi
66 −1226.73 −0.386429
77 −8707.31 −1.37070 −0.685351 0.728213i 0.740351π-0.740351\pi
−0.685351 + 0.728213i 0.740351π0.740351\pi
88 10353.8 0.893705
99 −17183.3 −0.873001
1010 −44532.3 −1.40824
1111 −82919.7 −1.70762 −0.853809 0.520587i 0.825713π-0.825713\pi
−0.853809 + 0.520587i 0.825713π0.825713\pi
1212 4500.61 0.0626550
1313 −28561.0 −0.277350
1414 213643. 1.48632
1515 90743.8 0.462814
1616 −300130. −1.14490
1717 374907. 1.08869 0.544344 0.838862i 0.316779π-0.316779\pi
0.544344 + 0.838862i 0.316779π0.316779\pi
1818 421610. 0.946638
1919 −361501. −0.636383 −0.318191 0.948026i 0.603075π-0.603075\pi
−0.318191 + 0.948026i 0.603075π0.603075\pi
2020 163379. 0.228329
2121 −435341. −0.488476
2222 2.03452e6 1.85165
2323 −2.31340e6 −1.72376 −0.861878 0.507116i 0.830712π-0.830712\pi
−0.861878 + 0.507116i 0.830712π0.830712\pi
2424 517661. 0.318489
2525 1.34101e6 0.686599
2626 700774. 0.300745
2727 −1.84321e6 −0.667480
2828 −783807. −0.240989
2929 −649654. −0.170565 −0.0852827 0.996357i 0.527179π-0.527179\pi
−0.0852827 + 0.996357i 0.527179π0.527179\pi
3030 −2.22649e6 −0.501852
3131 4.32521e6 0.841162 0.420581 0.907255i 0.361826π-0.361826\pi
0.420581 + 0.907255i 0.361826π0.361826\pi
3232 2.06285e6 0.347771
3333 −4.14575e6 −0.608542
3434 −9.19873e6 −1.18052
3535 −1.58036e7 −1.78012
3636 −1.54679e6 −0.153486
3737 1.21021e7 1.06158 0.530790 0.847503i 0.321895π-0.321895\pi
0.530790 + 0.847503i 0.321895π0.321895\pi
3838 8.86981e6 0.690062
3939 −1.42797e6 −0.0988391
4040 1.87919e7 1.16065
4141 2.59960e6 0.143674 0.0718372 0.997416i 0.477114π-0.477114\pi
0.0718372 + 0.997416i 0.477114π0.477114\pi
4242 1.06816e7 0.529679
4343 3.08318e6 0.137528 0.0687640 0.997633i 0.478094π-0.478094\pi
0.0687640 + 0.997633i 0.478094π0.478094\pi
4444 −7.46419e6 −0.300224
4545 −3.11872e7 −1.13376
4646 5.67617e7 1.86915
4747 2.60298e7 0.778090 0.389045 0.921219i 0.372805π-0.372805\pi
0.389045 + 0.921219i 0.372805π0.372805\pi
4848 −1.50057e7 −0.408009
4949 3.54636e7 0.878822
5050 −3.29032e7 −0.744513
5151 1.87443e7 0.387975
5252 −2.57098e6 −0.0487622
5353 −1.01982e8 −1.77535 −0.887673 0.460475i 0.847679π-0.847679\pi
−0.887673 + 0.460475i 0.847679π0.847679\pi
5454 4.52251e7 0.723782
5555 −1.50497e8 −2.21767
5656 −9.01536e7 −1.22500
5757 −1.80741e7 −0.226787
5858 1.59399e7 0.184953
5959 1.37562e8 1.47797 0.738985 0.673722i 0.235305π-0.235305\pi
0.738985 + 0.673722i 0.235305π0.235305\pi
6060 8.16850e6 0.0813695
6161 −4.29579e7 −0.397245 −0.198623 0.980076i 0.563647π-0.563647\pi
−0.198623 + 0.980076i 0.563647π0.563647\pi
6262 −1.06124e8 −0.912114
6363 1.49620e8 1.19662
6464 1.03052e8 0.767798
6565 −5.18375e7 −0.360192
6666 1.01720e8 0.659873
6767 5.54479e7 0.336162 0.168081 0.985773i 0.446243π-0.446243\pi
0.168081 + 0.985773i 0.446243π0.446243\pi
6868 3.37480e7 0.191407
6969 −1.15664e8 −0.614294
7070 3.87757e8 1.93027
7171 −1.43613e8 −0.670706 −0.335353 0.942093i 0.608856π-0.608856\pi
−0.335353 + 0.942093i 0.608856π0.608856\pi
7272 −1.77912e8 −0.780205
7373 3.37147e7 0.138952 0.0694762 0.997584i 0.477867π-0.477867\pi
0.0694762 + 0.997584i 0.477867π0.477867\pi
7474 −2.96937e8 −1.15112
7575 6.70469e7 0.244683
7676 −3.25413e7 −0.111886
7777 7.22007e8 2.34063
7878 3.50367e7 0.107176
7979 −2.66652e8 −0.770236 −0.385118 0.922867i 0.625839π-0.625839\pi
−0.385118 + 0.922867i 0.625839π0.625839\pi
8080 −5.44728e8 −1.48688
8181 2.46063e8 0.635132
8282 −6.37839e7 −0.155793
8383 −9.89542e7 −0.228867 −0.114433 0.993431i 0.536505π-0.536505\pi
−0.114433 + 0.993431i 0.536505π0.536505\pi
8484 −3.91882e7 −0.0858812
8585 6.80447e8 1.41387
8686 −7.56490e7 −0.149128
8787 −3.24809e7 −0.0607843
8888 −8.58533e8 −1.52611
8989 −5.08803e8 −0.859597 −0.429798 0.902925i 0.641415π-0.641415\pi
−0.429798 + 0.902925i 0.641415π0.641415\pi
9090 7.65211e8 1.22939
9191 2.48689e8 0.380164
9292 −2.08246e8 −0.303062
9393 2.16249e8 0.299764
9494 −6.38667e8 −0.843722
9595 −6.56116e8 −0.826465
9696 1.03137e8 0.123935
9797 −4.53920e8 −0.520603 −0.260301 0.965527i 0.583822π-0.583822\pi
−0.260301 + 0.965527i 0.583822π0.583822\pi
9898 −8.70137e8 −0.952950
9999 1.42483e9 1.49075
100100 1.20714e8 0.120714
101101 6.44663e8 0.616434 0.308217 0.951316i 0.400268π-0.400268\pi
0.308217 + 0.951316i 0.400268π0.400268\pi
102102 −4.59911e8 −0.420700
103103 −1.14825e9 −1.00524 −0.502620 0.864507i 0.667631π-0.667631\pi
−0.502620 + 0.864507i 0.667631π0.667631\pi
104104 −2.95715e8 −0.247869
105105 −7.90134e8 −0.634379
106106 2.50224e9 1.92510
107107 5.95299e8 0.439044 0.219522 0.975607i 0.429550π-0.429550\pi
0.219522 + 0.975607i 0.429550π0.429550\pi
108108 −1.65921e8 −0.117353
109109 1.00898e9 0.684645 0.342322 0.939583i 0.388787π-0.388787\pi
0.342322 + 0.939583i 0.388787π0.388787\pi
110110 3.69261e9 2.40473
111111 6.05071e8 0.378315
112112 2.61332e9 1.56932
113113 −1.59931e9 −0.922739 −0.461370 0.887208i 0.652642π-0.652642\pi
−0.461370 + 0.887208i 0.652642π0.652642\pi
114114 4.43466e8 0.245917
115115 −4.19877e9 −2.23863
116116 −5.84800e7 −0.0299879
117117 4.90772e8 0.242127
118118 −3.37524e9 −1.60264
119119 −3.26443e9 −1.49226
120120 9.39542e8 0.413619
121121 4.51772e9 1.91596
122122 1.05402e9 0.430753
123123 1.29973e8 0.0512011
124124 3.89343e8 0.147889
125125 −1.11097e9 −0.407011
126126 −3.67108e9 −1.29756
127127 −1.58816e9 −0.541723 −0.270861 0.962618i 0.587308π-0.587308\pi
−0.270861 + 0.962618i 0.587308π0.587308\pi
128128 −3.58467e9 −1.18033
129129 1.54151e8 0.0490107
130130 1.27189e9 0.390574
131131 −3.54703e9 −1.05231 −0.526156 0.850388i 0.676367π-0.676367\pi
−0.526156 + 0.850388i 0.676367π0.676367\pi
132132 −3.73189e8 −0.106991
133133 3.14770e9 0.872291
134134 −1.36047e9 −0.364517
135135 −3.34539e9 −0.866850
136136 3.88170e9 0.972965
137137 −3.79917e9 −0.921395 −0.460697 0.887557i 0.652401π-0.652401\pi
−0.460697 + 0.887557i 0.652401π0.652401\pi
138138 2.83793e9 0.666109
139139 3.81526e9 0.866878 0.433439 0.901183i 0.357300π-0.357300\pi
0.433439 + 0.901183i 0.357300π0.357300\pi
140140 −1.42259e9 −0.312971
141141 1.30142e9 0.277287
142142 3.52370e9 0.727280
143143 2.36827e9 0.473608
144144 5.15721e9 0.999502
145145 −1.17911e9 −0.221512
146146 −8.27225e8 −0.150673
147147 1.77308e9 0.313185
148148 1.08940e9 0.186641
149149 1.16149e10 1.93054 0.965269 0.261258i 0.0841374π-0.0841374\pi
0.965269 + 0.261258i 0.0841374π0.0841374\pi
150150 −1.64507e9 −0.265322
151151 3.19196e9 0.499645 0.249822 0.968292i 0.419628π-0.419628\pi
0.249822 + 0.968292i 0.419628π0.419628\pi
152152 −3.74291e9 −0.568739
153153 −6.44213e9 −0.950425
154154 −1.77152e10 −2.53806
155155 7.85016e9 1.09241
156156 −1.28542e8 −0.0173774
157157 −5.05861e9 −0.664481 −0.332240 0.943195i 0.607805π-0.607805\pi
−0.332240 + 0.943195i 0.607805π0.607805\pi
158158 6.54260e9 0.835205
159159 −5.09883e9 −0.632679
160160 3.74403e9 0.451647
161161 2.01435e10 2.36275
162162 −6.03741e9 −0.688705
163163 −7.83438e9 −0.869282 −0.434641 0.900604i 0.643125π-0.643125\pi
−0.434641 + 0.900604i 0.643125π0.643125\pi
164164 2.34009e8 0.0252601
165165 −7.52444e9 −0.790308
166166 2.42794e9 0.248172
167167 −1.44048e8 −0.0143312 −0.00716559 0.999974i 0.502281π-0.502281\pi
−0.00716559 + 0.999974i 0.502281π0.502281\pi
168168 −4.50743e9 −0.436553
169169 8.15731e8 0.0769231
170170 −1.66955e10 −1.53313
171171 6.21178e9 0.555563
172172 2.77539e8 0.0241794
173173 −1.36123e10 −1.15538 −0.577689 0.816257i 0.696045π-0.696045\pi
−0.577689 + 0.816257i 0.696045π0.696045\pi
174174 7.96952e8 0.0659114
175175 −1.16766e10 −0.941122
176176 2.48867e10 1.95506
177177 6.87774e9 0.526703
178178 1.24840e10 0.932104
179179 4.98195e8 0.0362711 0.0181355 0.999836i 0.494227π-0.494227\pi
0.0181355 + 0.999836i 0.494227π0.494227\pi
180180 −2.80739e9 −0.199331
181181 2.86758e9 0.198592 0.0992961 0.995058i 0.468341π-0.468341\pi
0.0992961 + 0.995058i 0.468341π0.468341\pi
182182 −6.10185e9 −0.412231
183183 −2.14777e9 −0.141566
184184 −2.39525e10 −1.54053
185185 2.19650e10 1.37866
186186 −5.30588e9 −0.325049
187187 −3.10871e10 −1.85906
188188 2.34312e9 0.136800
189189 1.60494e10 0.914916
190190 1.60985e10 0.896177
191191 −2.83331e10 −1.54044 −0.770218 0.637781i 0.779852π-0.779852\pi
−0.770218 + 0.637781i 0.779852π0.779852\pi
192192 5.15232e9 0.273620
193193 2.44273e10 1.26726 0.633632 0.773634i 0.281563π-0.281563\pi
0.633632 + 0.773634i 0.281563π0.281563\pi
194194 1.11374e10 0.564516
195195 −2.59173e9 −0.128361
196196 3.19233e9 0.154510
197197 1.37323e10 0.649601 0.324800 0.945783i 0.394703π-0.394703\pi
0.324800 + 0.945783i 0.394703π0.394703\pi
198198 −3.49597e10 −1.61650
199199 −8.56614e9 −0.387210 −0.193605 0.981080i 0.562018π-0.562018\pi
−0.193605 + 0.981080i 0.562018π0.562018\pi
200200 1.38846e10 0.613617
201201 2.77224e9 0.119798
202202 −1.58175e10 −0.668430
203203 5.65673e9 0.233794
204204 1.68731e9 0.0682117
205205 4.71821e9 0.186589
206206 2.81736e10 1.09003
207207 3.97518e10 1.50484
208208 8.57200e9 0.317539
209209 2.99756e10 1.08670
210210 1.93868e10 0.687889
211211 2.70272e9 0.0938706 0.0469353 0.998898i 0.485055π-0.485055\pi
0.0469353 + 0.998898i 0.485055π0.485055\pi
212212 −9.18015e9 −0.312132
213213 −7.18027e9 −0.239019
214214 −1.46063e10 −0.476078
215215 5.59590e9 0.178606
216216 −1.90842e10 −0.596530
217217 −3.76610e10 −1.15298
218218 −2.47565e10 −0.742394
219219 1.68564e9 0.0495184
220220 −1.35473e10 −0.389899
221221 −1.07077e10 −0.301947
222222 −1.48460e10 −0.410225
223223 −3.57969e10 −0.969333 −0.484667 0.874699i 0.661059π-0.661059\pi
−0.484667 + 0.874699i 0.661059π0.661059\pi
224224 −1.79619e10 −0.476690
225225 −2.30430e10 −0.599401
226226 3.92407e10 1.00057
227227 −7.55373e10 −1.88819 −0.944094 0.329675i 0.893061π-0.893061\pi
−0.944094 + 0.329675i 0.893061π0.893061\pi
228228 −1.62698e9 −0.0398726
229229 −5.69066e10 −1.36742 −0.683711 0.729753i 0.739635π-0.739635\pi
−0.683711 + 0.729753i 0.739635π0.739635\pi
230230 1.03021e11 2.42745
231231 3.60984e10 0.834130
232232 −6.72637e9 −0.152435
233233 −4.68000e10 −1.04026 −0.520132 0.854086i 0.674117π-0.674117\pi
−0.520132 + 0.854086i 0.674117π0.674117\pi
234234 −1.20416e10 −0.262550
235235 4.72434e10 1.01050
236236 1.23830e10 0.259849
237237 −1.33319e10 −0.274488
238238 8.00961e10 1.61814
239239 −8.79993e9 −0.174457 −0.0872285 0.996188i 0.527801π-0.527801\pi
−0.0872285 + 0.996188i 0.527801π0.527801\pi
240240 −2.72349e10 −0.529877
241241 −1.76057e10 −0.336183 −0.168092 0.985771i 0.553760π-0.553760\pi
−0.168092 + 0.985771i 0.553760π0.553760\pi
242242 −1.10847e11 −2.07757
243243 4.85824e10 0.893821
244244 −3.86695e9 −0.0698415
245245 6.43656e10 1.14132
246246 −3.18902e9 −0.0555200
247247 1.03248e10 0.176501
248248 4.47823e10 0.751751
249249 −4.94743e9 −0.0815611
250250 2.72588e10 0.441343
251251 −1.77113e10 −0.281655 −0.140828 0.990034i 0.544976π-0.544976\pi
−0.140828 + 0.990034i 0.544976π0.544976\pi
252252 1.34684e10 0.210384
253253 1.91826e11 2.94351
254254 3.89671e10 0.587417
255255 3.40204e10 0.503859
256256 3.51910e10 0.512096
257257 −6.26733e10 −0.896156 −0.448078 0.893994i 0.647891π-0.647891\pi
−0.448078 + 0.893994i 0.647891π0.647891\pi
258258 −3.78224e9 −0.0531448
259259 −1.05377e11 −1.45511
260260 −4.66627e9 −0.0633271
261261 1.11632e10 0.148904
262262 8.70301e10 1.14107
263263 4.49501e10 0.579335 0.289667 0.957127i 0.406455π-0.406455\pi
0.289667 + 0.957127i 0.406455π0.406455\pi
264264 −4.29243e10 −0.543857
265265 −1.85095e11 −2.30563
266266 −7.72322e10 −0.945869
267267 −2.54388e10 −0.306334
268268 4.99126e9 0.0591023
269269 −8.65536e10 −1.00786 −0.503930 0.863745i 0.668113π-0.668113\pi
−0.503930 + 0.863745i 0.668113π0.668113\pi
270270 8.20825e10 0.939969
271271 8.36056e10 0.941615 0.470808 0.882236i 0.343963π-0.343963\pi
0.470808 + 0.882236i 0.343963π0.343963\pi
272272 −1.12521e11 −1.24644
273273 1.24338e10 0.135479
274274 9.32165e10 0.999114
275275 −1.11196e11 −1.17245
276276 −1.04117e10 −0.108002
277277 1.73137e11 1.76698 0.883489 0.468452i 0.155188π-0.155188\pi
0.883489 + 0.468452i 0.155188π0.155188\pi
278278 −9.36114e10 −0.939999
279279 −7.43213e10 −0.734335
280280 −1.63627e11 −1.59090
281281 7.57598e10 0.724871 0.362435 0.932009i 0.381945π-0.381945\pi
0.362435 + 0.932009i 0.381945π0.381945\pi
282282 −3.19316e10 −0.300676
283283 1.68901e11 1.56528 0.782642 0.622472i 0.213872π-0.213872\pi
0.782642 + 0.622472i 0.213872π0.213872\pi
284284 −1.29277e10 −0.117920
285285 −3.28040e10 −0.294527
286286 −5.81079e10 −0.513557
287287 −2.26355e10 −0.196935
288288 −3.54466e10 −0.303605
289289 2.19672e10 0.185240
290290 2.89306e10 0.240196
291291 −2.26947e10 −0.185527
292292 3.03490e9 0.0244299
293293 −1.18842e11 −0.942030 −0.471015 0.882125i 0.656112π-0.656112\pi
−0.471015 + 0.882125i 0.656112π0.656112\pi
294294 −4.35044e10 −0.339602
295295 2.49672e11 1.91943
296296 1.25302e11 0.948740
297297 1.52839e11 1.13980
298298 −2.84984e11 −2.09338
299299 6.60730e10 0.478084
300300 6.03537e9 0.0430188
301301 −2.68462e10 −0.188510
302302 −7.83181e10 −0.541790
303303 3.22313e10 0.219678
304304 1.08497e11 0.728597
305305 −7.79675e10 −0.515899
306306 1.58064e11 1.03059
307307 −1.99530e11 −1.28199 −0.640997 0.767543i 0.721479π-0.721479\pi
−0.640997 + 0.767543i 0.721479π0.721479\pi
308308 6.49930e10 0.411518
309309 −5.74095e10 −0.358237
310310 −1.92612e11 −1.18455
311311 1.90145e11 1.15256 0.576281 0.817252i 0.304503π-0.304503\pi
0.576281 + 0.817252i 0.304503π0.304503\pi
312312 −1.47849e10 −0.0883330
313313 −3.39674e10 −0.200038 −0.100019 0.994986i 0.531890π-0.531890\pi
−0.100019 + 0.994986i 0.531890π0.531890\pi
314314 1.24118e11 0.720530
315315 2.71557e11 1.55404
316316 −2.40033e10 −0.135419
317317 −8.53565e9 −0.0474756 −0.0237378 0.999718i 0.507557π-0.507557\pi
−0.0237378 + 0.999718i 0.507557π0.507557\pi
318318 1.25105e11 0.686045
319319 5.38691e10 0.291260
320320 1.87037e11 0.997133
321321 2.97633e10 0.156462
322322 −4.94242e11 −2.56205
323323 −1.35529e11 −0.692822
324324 2.21499e10 0.111665
325325 −3.83007e10 −0.190428
326326 1.92225e11 0.942606
327327 5.04464e10 0.243986
328328 2.69157e10 0.128403
329329 −2.26649e11 −1.06653
330330 1.84620e11 0.856971
331331 5.09528e10 0.233315 0.116657 0.993172i 0.462782π-0.462782\pi
0.116657 + 0.993172i 0.462782π0.462782\pi
332332 −8.90757e9 −0.0402381
333333 −2.07954e11 −0.926760
334334 3.53436e9 0.0155400
335335 1.00637e11 0.436571
336336 1.30659e11 0.559258
337337 2.31891e11 0.979375 0.489688 0.871898i 0.337111π-0.337111\pi
0.489688 + 0.871898i 0.337111π0.337111\pi
338338 −2.00148e10 −0.0834115
339339 −7.99609e10 −0.328836
340340 6.12519e10 0.248579
341341 −3.58645e11 −1.43638
342342 −1.52412e11 −0.602425
343343 4.25786e10 0.166099
344344 3.19226e10 0.122909
345345 −2.09927e11 −0.797778
346346 3.33992e11 1.25283
347347 −3.64138e10 −0.134829 −0.0674146 0.997725i 0.521475π-0.521475\pi
−0.0674146 + 0.997725i 0.521475π0.521475\pi
348348 −2.92384e9 −0.0106868
349349 9.86839e10 0.356067 0.178034 0.984024i 0.443026π-0.443026\pi
0.178034 + 0.984024i 0.443026π0.443026\pi
350350 2.86498e11 1.02051
351351 5.26440e10 0.185126
352352 −1.71051e11 −0.593860
353353 −3.90458e11 −1.33841 −0.669204 0.743078i 0.733365π-0.733365\pi
−0.669204 + 0.743078i 0.733365π0.733365\pi
354354 −1.68752e11 −0.571131
355355 −2.60655e11 −0.871040
356356 −4.58010e10 −0.151130
357357 −1.63212e11 −0.531797
358358 −1.22237e10 −0.0393305
359359 −3.86316e11 −1.22749 −0.613744 0.789505i 0.710337π-0.710337\pi
−0.613744 + 0.789505i 0.710337π0.710337\pi
360360 −3.22906e11 −1.01325
361361 −1.92005e11 −0.595017
362362 −7.03591e10 −0.215343
363363 2.25874e11 0.682788
364364 2.23863e10 0.0668385
365365 6.11914e10 0.180456
366366 5.26979e10 0.153507
367367 1.48155e11 0.426305 0.213152 0.977019i 0.431627π-0.431627\pi
0.213152 + 0.977019i 0.431627π0.431627\pi
368368 6.94320e11 1.97353
369369 −4.46697e10 −0.125428
370370 −5.38934e11 −1.49495
371371 8.87991e11 2.43347
372372 1.94661e10 0.0527030
373373 −6.15338e11 −1.64598 −0.822989 0.568057i 0.807695π-0.807695\pi
−0.822989 + 0.568057i 0.807695π0.807695\pi
374374 7.62755e11 2.01587
375375 −5.55453e10 −0.145046
376376 2.69507e11 0.695383
377377 1.85548e10 0.0473063
378378 −3.93789e11 −0.992089
379379 2.48822e10 0.0619459 0.0309729 0.999520i 0.490139π-0.490139\pi
0.0309729 + 0.999520i 0.490139π0.490139\pi
380380 −5.90617e10 −0.145305
381381 −7.94035e10 −0.193053
382382 6.95181e11 1.67037
383383 4.82478e11 1.14573 0.572866 0.819649i 0.305832π-0.305832\pi
0.572866 + 0.819649i 0.305832π0.305832\pi
384384 −1.79224e11 −0.420634
385385 1.31043e12 3.03976
386386 −5.99349e11 −1.37416
387387 −5.29792e10 −0.120062
388388 −4.08606e10 −0.0915297
389389 3.80215e11 0.841890 0.420945 0.907086i 0.361698π-0.361698\pi
0.420945 + 0.907086i 0.361698π0.361698\pi
390390 6.35909e10 0.139189
391391 −8.67310e11 −1.87663
392392 3.67183e11 0.785408
393393 −1.77342e11 −0.375012
394394 −3.36937e11 −0.704394
395395 −4.83968e11 −1.00030
396396 1.28259e11 0.262096
397397 7.91759e11 1.59969 0.799845 0.600207i 0.204915π-0.204915\pi
0.799845 + 0.600207i 0.204915π0.204915\pi
398398 2.10179e11 0.419871
399399 1.57376e11 0.310858
400400 −4.02478e11 −0.786090
401401 −3.06132e11 −0.591234 −0.295617 0.955307i 0.595525π-0.595525\pi
−0.295617 + 0.955307i 0.595525π0.595525\pi
402402 −6.80199e10 −0.129903
403403 −1.23532e11 −0.233296
404404 5.80307e10 0.108378
405405 4.46598e11 0.824840
406406 −1.38794e11 −0.253515
407407 −1.00350e12 −1.81277
408408 1.94074e11 0.346735
409409 9.04916e11 1.59902 0.799509 0.600654i 0.205093π-0.205093\pi
0.799509 + 0.600654i 0.205093π0.205093\pi
410410 −1.15766e11 −0.202327
411411 −1.89948e11 −0.328357
412412 −1.03362e11 −0.176736
413413 −1.19780e12 −2.02586
414414 −9.75352e11 −1.63177
415415 −1.79599e11 −0.297227
416416 −5.89172e10 −0.0964544
417417 1.90753e11 0.308929
418418 −7.35482e11 −1.17836
419419 3.36210e11 0.532903 0.266451 0.963848i 0.414149π-0.414149\pi
0.266451 + 0.963848i 0.414149π0.414149\pi
420420 −7.11256e10 −0.111533
421421 −1.87100e11 −0.290271 −0.145135 0.989412i 0.546362π-0.546362\pi
−0.145135 + 0.989412i 0.546362π0.546362\pi
422422 −6.63140e10 −0.101789
423423 −4.47276e11 −0.679273
424424 −1.05590e12 −1.58664
425425 5.02755e11 0.747491
426426 1.76175e11 0.259180
427427 3.74048e11 0.544504
428428 5.35872e10 0.0771905
429429 1.18407e11 0.168779
430430 −1.37301e11 −0.193672
431431 9.60500e11 1.34076 0.670378 0.742020i 0.266132π-0.266132\pi
0.670378 + 0.742020i 0.266132π0.266132\pi
432432 5.53203e11 0.764200
433433 1.14812e12 1.56961 0.784806 0.619741i 0.212762π-0.212762\pi
0.784806 + 0.619741i 0.212762π0.212762\pi
434434 9.24051e11 1.25024
435435 −5.89520e10 −0.0789400
436436 9.08259e10 0.120371
437437 8.36297e11 1.09697
438438 −4.13590e10 −0.0536953
439439 5.44277e11 0.699407 0.349703 0.936860i 0.386282π-0.386282\pi
0.349703 + 0.936860i 0.386282π0.386282\pi
440440 −1.55822e12 −1.98194
441441 −6.09381e11 −0.767212
442442 2.62725e11 0.327417
443443 −1.12065e11 −0.138247 −0.0691233 0.997608i 0.522020π-0.522020\pi
−0.0691233 + 0.997608i 0.522020π0.522020\pi
444444 5.44668e10 0.0665133
445445 −9.23466e11 −1.11635
446446 8.78313e11 1.05110
447447 5.80714e11 0.687984
448448 −8.97307e11 −1.05242
449449 −1.11883e12 −1.29913 −0.649567 0.760304i 0.725050π-0.725050\pi
−0.649567 + 0.760304i 0.725050π0.725050\pi
450450 5.65384e11 0.649961
451451 −2.15558e11 −0.245341
452452 −1.43965e11 −0.162231
453453 1.59589e11 0.178058
454454 1.85339e12 2.04746
455455 4.51365e11 0.493716
456456 −1.87135e11 −0.202681
457457 −1.38914e12 −1.48978 −0.744890 0.667188i 0.767498π-0.767498\pi
−0.744890 + 0.667188i 0.767498π0.767498\pi
458458 1.39626e12 1.48276
459459 −6.91032e11 −0.726677
460460 −3.77961e11 −0.393583
461461 1.06041e12 1.09350 0.546752 0.837294i 0.315864π-0.315864\pi
0.546752 + 0.837294i 0.315864π0.315864\pi
462462 −8.85711e11 −0.904488
463463 2.81576e11 0.284762 0.142381 0.989812i 0.454524π-0.454524\pi
0.142381 + 0.989812i 0.454524π0.454524\pi
464464 1.94980e11 0.195281
465465 3.92486e11 0.389301
466466 1.14829e12 1.12801
467467 −9.91125e11 −0.964278 −0.482139 0.876095i 0.660140π-0.660140\pi
−0.482139 + 0.876095i 0.660140π0.660140\pi
468468 4.41779e10 0.0425695
469469 −4.82802e11 −0.460778
470470 −1.15917e12 −1.09573
471471 −2.52916e11 −0.236801
472472 1.42429e12 1.32087
473473 −2.55656e11 −0.234845
474474 3.27112e11 0.297641
475475 −4.84778e11 −0.436940
476476 −2.93855e11 −0.262362
477477 1.75239e12 1.54988
478478 2.15915e11 0.189173
479479 −1.38074e12 −1.19841 −0.599203 0.800597i 0.704516π-0.704516\pi
−0.599203 + 0.800597i 0.704516π0.704516\pi
480480 1.87191e11 0.160953
481481 −3.45648e11 −0.294429
482482 4.31974e11 0.364541
483483 1.00712e12 0.842013
484484 4.06673e11 0.336853
485485 −8.23854e11 −0.676102
486486 −1.19202e12 −0.969215
487487 9.65212e11 0.777575 0.388788 0.921327i 0.372894π-0.372894\pi
0.388788 + 0.921327i 0.372894π0.372894\pi
488488 −4.44777e11 −0.355020
489489 −3.91697e11 −0.309785
490490 −1.57928e12 −1.23759
491491 −1.10735e12 −0.859841 −0.429920 0.902867i 0.641458π-0.641458\pi
−0.429920 + 0.902867i 0.641458π0.641458\pi
492492 1.16998e10 0.00900191
493493 −2.43559e11 −0.185692
494494 −2.53331e11 −0.191389
495495 2.58604e12 1.93603
496496 −1.29812e12 −0.963050
497497 1.25049e12 0.919338
498498 1.21390e11 0.0884407
499499 −2.24583e12 −1.62153 −0.810763 0.585374i 0.800948π-0.800948\pi
−0.810763 + 0.585374i 0.800948π0.800948\pi
500500 −1.00006e11 −0.0715586
501501 −7.20198e9 −0.00510719
502502 4.34564e11 0.305413
503503 3.98259e11 0.277402 0.138701 0.990334i 0.455707π-0.455707\pi
0.138701 + 0.990334i 0.455707π0.455707\pi
504504 1.54913e12 1.06943
505505 1.17005e12 0.800557
506506 −4.70666e12 −3.19180
507507 4.07843e10 0.0274130
508508 −1.42961e11 −0.0952428
509509 2.66411e12 1.75923 0.879615 0.475687i 0.157800π-0.157800\pi
0.879615 + 0.475687i 0.157800π0.157800\pi
510510 −8.34727e11 −0.546360
511511 −2.93564e11 −0.190462
512512 9.71905e11 0.625042
513513 6.66323e11 0.424773
514514 1.53775e12 0.971747
515515 −2.08405e12 −1.30550
516516 1.38762e10 0.00861681
517517 −2.15838e12 −1.32868
518518 2.58553e12 1.57785
519519 −6.80577e11 −0.411741
520520 −5.36715e11 −0.321906
521521 −3.11961e12 −1.85494 −0.927472 0.373894i 0.878022π-0.878022\pi
−0.927472 + 0.373894i 0.878022π0.878022\pi
522522 −2.73900e11 −0.161464
523523 −2.05566e12 −1.20142 −0.600709 0.799468i 0.705115π-0.705115\pi
−0.600709 + 0.799468i 0.705115π0.705115\pi
524524 −3.19294e11 −0.185012
525525 −5.83798e11 −0.335387
526526 −1.10290e12 −0.628202
527527 1.62155e12 0.915762
528528 1.24426e12 0.696722
529529 3.55067e12 1.97133
530530 4.54150e12 2.50011
531531 −2.36377e12 −1.29027
532532 2.83347e11 0.153362
533533 −7.42472e10 −0.0398481
534534 6.24166e11 0.332173
535535 1.08045e12 0.570183
536536 5.74096e11 0.300430
537537 2.49083e10 0.0129259
538538 2.12368e12 1.09287
539539 −2.94063e12 −1.50069
540540 −3.01142e11 −0.152405
541541 8.06876e11 0.404966 0.202483 0.979286i 0.435099π-0.435099\pi
0.202483 + 0.979286i 0.435099π0.435099\pi
542542 −2.05135e12 −1.02104
543543 1.43371e11 0.0707721
544544 7.73378e11 0.378614
545545 1.83128e12 0.889142
546546 −3.05076e11 −0.146906
547547 1.93370e12 0.923521 0.461760 0.887005i 0.347218π-0.347218\pi
0.461760 + 0.887005i 0.347218π0.347218\pi
548548 −3.41990e11 −0.161995
549549 7.38157e11 0.346795
550550 2.72832e12 1.27134
551551 2.34851e11 0.108545
552552 −1.19756e12 −0.548997
553553 2.32183e12 1.05576
554554 −4.24810e12 −1.91602
555555 1.09819e12 0.491314
556556 3.43439e11 0.152410
557557 3.06623e12 1.34976 0.674879 0.737928i 0.264196π-0.264196\pi
0.674879 + 0.737928i 0.264196π0.264196\pi
558558 1.82355e12 0.796276
559559 −8.80587e10 −0.0381434
560560 4.74312e12 2.03806
561561 −1.55427e12 −0.662512
562562 −1.85885e12 −0.786013
563563 −2.59944e12 −1.09042 −0.545208 0.838301i 0.683549π-0.683549\pi
−0.545208 + 0.838301i 0.683549π0.683549\pi
564564 1.17150e11 0.0487512
565565 −2.90270e12 −1.19835
566566 −4.14416e12 −1.69732
567567 −2.14255e12 −0.870576
568568 −1.48694e12 −0.599413
569569 −1.63182e12 −0.652632 −0.326316 0.945261i 0.605807π-0.605807\pi
−0.326316 + 0.945261i 0.605807π0.605807\pi
570570 8.04880e11 0.319370
571571 −1.05673e12 −0.416010 −0.208005 0.978128i 0.566697π-0.566697\pi
−0.208005 + 0.978128i 0.566697π0.566697\pi
572572 2.13185e11 0.0832672
573573 −1.41658e12 −0.548964
574574 5.55386e11 0.213546
575575 −3.10230e12 −1.18353
576576 −1.77077e12 −0.670289
577577 −8.51110e11 −0.319665 −0.159832 0.987144i 0.551095π-0.551095\pi
−0.159832 + 0.987144i 0.551095π0.551095\pi
578578 −5.38987e11 −0.200865
579579 1.22130e12 0.451614
580580 −1.06140e11 −0.0389450
581581 8.61624e11 0.313708
582582 5.56839e11 0.201176
583583 8.45633e12 3.03161
584584 3.49075e11 0.124183
585585 8.90739e11 0.314448
586586 2.91591e12 1.02149
587587 6.19927e11 0.215511 0.107755 0.994177i 0.465634π-0.465634\pi
0.107755 + 0.994177i 0.465634π0.465634\pi
588588 1.59608e11 0.0550626
589589 −1.56357e12 −0.535301
590590 −6.12597e12 −2.08133
591591 6.86579e11 0.231498
592592 −3.63220e12 −1.21541
593593 −3.79034e12 −1.25873 −0.629364 0.777110i 0.716685π-0.716685\pi
−0.629364 + 0.777110i 0.716685π0.716685\pi
594594 −3.75005e12 −1.23594
595595 −5.92486e12 −1.93799
596596 1.04554e12 0.339417
597597 −4.28283e11 −0.137990
598598 −1.62117e12 −0.518410
599599 2.62045e12 0.831679 0.415839 0.909438i 0.363488π-0.363488\pi
0.415839 + 0.909438i 0.363488π0.363488\pi
600600 6.94190e11 0.218674
601601 1.11042e12 0.347178 0.173589 0.984818i 0.444464π-0.444464\pi
0.173589 + 0.984818i 0.444464π0.444464\pi
602602 6.58700e11 0.204411
603603 −9.52777e11 −0.293470
604604 2.87331e11 0.0878449
605605 8.19956e12 2.48824
606606 −7.90829e11 −0.238208
607607 4.57077e12 1.36660 0.683298 0.730140i 0.260545π-0.260545\pi
0.683298 + 0.730140i 0.260545π0.260545\pi
608608 −7.45724e11 −0.221316
609609 2.82821e11 0.0833171
610610 1.91301e12 0.559415
611611 −7.43436e11 −0.215803
612612 −5.79902e11 −0.167099
613613 −5.61998e12 −1.60754 −0.803772 0.594938i 0.797177π-0.797177\pi
−0.803772 + 0.594938i 0.797177π0.797177\pi
614614 4.89568e12 1.39013
615615 2.35898e11 0.0664945
616616 7.47551e12 2.09184
617617 2.17420e12 0.603970 0.301985 0.953313i 0.402351π-0.402351\pi
0.301985 + 0.953313i 0.402351π0.402351\pi
618618 1.40860e12 0.388454
619619 5.91455e12 1.61925 0.809625 0.586947i 0.199670π-0.199670\pi
0.809625 + 0.586947i 0.199670π0.199670\pi
620620 7.06649e11 0.192062
621621 4.26409e12 1.15057
622622 −4.66542e12 −1.24978
623623 4.43031e12 1.17825
624624 4.28576e11 0.113161
625625 −4.63555e12 −1.21518
626626 8.33426e11 0.216912
627627 1.49869e12 0.387266
628628 −4.55361e11 −0.116826
629629 4.53716e12 1.15573
630630 −6.66293e12 −1.68513
631631 −5.05658e12 −1.26977 −0.634884 0.772607i 0.718952π-0.718952\pi
−0.634884 + 0.772607i 0.718952π0.718952\pi
632632 −2.76086e12 −0.688364
633633 1.35128e11 0.0334526
634634 2.09431e11 0.0514801
635635 −2.88247e12 −0.703530
636636 −4.58982e11 −0.111234
637637 −1.01288e12 −0.243741
638638 −1.32173e12 −0.315828
639639 2.46775e12 0.585527
640640 −6.50609e12 −1.53289
641641 1.51370e12 0.354142 0.177071 0.984198i 0.443338π-0.443338\pi
0.177071 + 0.984198i 0.443338π0.443338\pi
642642 −7.30274e11 −0.169660
643643 3.90178e12 0.900146 0.450073 0.892992i 0.351398π-0.351398\pi
0.450073 + 0.892992i 0.351398π0.351398\pi
644644 1.81326e12 0.415407
645645 2.79779e11 0.0636498
646646 3.32535e12 0.751261
647647 7.88115e12 1.76815 0.884077 0.467341i 0.154788π-0.154788\pi
0.884077 + 0.467341i 0.154788π0.154788\pi
648648 2.54768e12 0.567620
649649 −1.14066e13 −2.52381
650650 9.39747e11 0.206491
651651 −1.88294e12 −0.410887
652652 −7.05229e11 −0.152833
653653 −5.10697e12 −1.09914 −0.549571 0.835447i 0.685209π-0.685209\pi
−0.549571 + 0.835447i 0.685209π0.685209\pi
654654 −1.23776e12 −0.264567
655655 −6.43778e12 −1.36663
656656 −7.80217e11 −0.164493
657657 −5.79329e11 −0.121306
658658 5.56107e12 1.15649
659659 −5.45308e11 −0.112631 −0.0563154 0.998413i 0.517935π-0.517935\pi
−0.0563154 + 0.998413i 0.517935π0.517935\pi
660660 −6.77329e11 −0.138948
661661 −3.05503e12 −0.622456 −0.311228 0.950335i 0.600740π-0.600740\pi
−0.311228 + 0.950335i 0.600740π0.600740\pi
662662 −1.25018e12 −0.252995
663663 −5.35356e11 −0.107605
664664 −1.02455e12 −0.204539
665665 5.71301e12 1.13284
666666 5.10236e12 1.00493
667667 1.50291e12 0.294013
668668 −1.29668e10 −0.00251963
669669 −1.78974e12 −0.345441
670670 −2.46923e12 −0.473395
671671 3.56205e12 0.678343
672672 −8.98046e11 −0.169878
673673 −1.57419e12 −0.295795 −0.147897 0.989003i 0.547251π-0.547251\pi
−0.147897 + 0.989003i 0.547251π0.547251\pi
674674 −5.68968e12 −1.06199
675675 −2.47177e12 −0.458291
676676 7.34298e10 0.0135242
677677 −2.62249e11 −0.0479805 −0.0239903 0.999712i 0.507637π-0.507637\pi
−0.0239903 + 0.999712i 0.507637π0.507637\pi
678678 1.96192e12 0.356573
679679 3.95242e12 0.713591
680680 7.04520e12 1.26358
681681 −3.77666e12 −0.672892
682682 8.79973e12 1.55754
683683 −3.64995e12 −0.641791 −0.320896 0.947115i 0.603984π-0.603984\pi
−0.320896 + 0.947115i 0.603984π0.603984\pi
684684 5.59166e11 0.0976762
685685 −6.89539e12 −1.19661
686686 −1.04471e12 −0.180110
687687 −2.84517e12 −0.487307
688688 −9.25354e11 −0.157456
689689 2.91271e12 0.492392
690690 5.15077e12 0.865070
691691 −1.69603e12 −0.282998 −0.141499 0.989938i 0.545192π-0.545192\pi
−0.141499 + 0.989938i 0.545192π0.545192\pi
692692 −1.22534e12 −0.203132
693693 −1.24065e13 −2.04337
694694 8.93451e11 0.146202
695695 6.92461e12 1.12581
696696 −3.36300e11 −0.0543232
697697 9.74608e11 0.156416
698698 −2.42131e12 −0.386101
699699 −2.33987e12 −0.370718
700700 −1.05110e12 −0.165463
701701 1.15906e13 1.81291 0.906454 0.422304i 0.138778π-0.138778\pi
0.906454 + 0.422304i 0.138778π0.138778\pi
702702 −1.29167e12 −0.200741
703703 −4.37492e12 −0.675571
704704 −8.54505e12 −1.31111
705705 2.36204e12 0.360111
706706 9.58030e12 1.45130
707707 −5.61328e12 −0.844946
708708 6.19114e11 0.0926022
709709 −6.98541e12 −1.03821 −0.519104 0.854711i 0.673734π-0.673734\pi
−0.519104 + 0.854711i 0.673734π0.673734\pi
710710 6.39544e12 0.944512
711711 4.58196e12 0.672417
712712 −5.26804e12 −0.768226
713713 −1.00059e13 −1.44996
714714 4.00459e12 0.576654
715715 4.29835e12 0.615070
716716 4.48461e10 0.00637699
717717 −4.39972e11 −0.0621711
718718 9.47866e12 1.33103
719719 −3.43730e12 −0.479665 −0.239832 0.970814i 0.577092π-0.577092\pi
−0.239832 + 0.970814i 0.577092π0.577092\pi
720720 9.36022e12 1.29804
721721 9.99820e12 1.37788
722722 4.71103e12 0.645206
723723 −8.80236e11 −0.119805
724724 2.58131e11 0.0349154
725725 −8.71194e11 −0.117110
726726 −5.54205e12 −0.740381
727727 1.25723e12 0.166920 0.0834601 0.996511i 0.473403π-0.473403\pi
0.0834601 + 0.996511i 0.473403π0.473403\pi
728728 2.57488e12 0.339755
729729 −2.41427e12 −0.316601
730730 −1.50139e12 −0.195678
731731 1.15591e12 0.149725
732732 −1.93337e11 −0.0248894
733733 −1.51950e12 −0.194416 −0.0972079 0.995264i 0.530991π-0.530991\pi
−0.0972079 + 0.995264i 0.530991π0.530991\pi
734734 −3.63515e12 −0.462264
735735 3.21810e12 0.406731
736736 −4.77221e12 −0.599473
737737 −4.59772e12 −0.574036
738738 1.09602e12 0.136008
739739 1.19661e13 1.47589 0.737944 0.674862i 0.235797π-0.235797\pi
0.737944 + 0.674862i 0.235797π0.235797\pi
740740 1.97723e12 0.242390
741741 5.16213e11 0.0628995
742742 −2.17878e13 −2.63873
743743 1.52097e12 0.183093 0.0915465 0.995801i 0.470819π-0.470819\pi
0.0915465 + 0.995801i 0.470819π0.470819\pi
744744 2.23899e12 0.267901
745745 2.10808e13 2.50717
746746 1.50980e13 1.78482
747747 1.70036e12 0.199801
748748 −2.79838e12 −0.326850
749749 −5.18346e12 −0.601799
750750 1.36286e12 0.157281
751751 −8.06595e12 −0.925286 −0.462643 0.886545i 0.653099π-0.653099\pi
−0.462643 + 0.886545i 0.653099π0.653099\pi
752752 −7.81230e12 −0.890838
753753 −8.85513e11 −0.100373
754754 −4.55260e11 −0.0512966
755755 5.79333e12 0.648884
756756 1.44472e12 0.160856
757757 8.72882e12 0.966104 0.483052 0.875592i 0.339528π-0.339528\pi
0.483052 + 0.875592i 0.339528π0.339528\pi
758758 −6.10511e11 −0.0671710
759759 9.59079e12 1.04898
760760 −6.79329e12 −0.738616
761761 7.63407e11 0.0825135 0.0412568 0.999149i 0.486864π-0.486864\pi
0.0412568 + 0.999149i 0.486864π0.486864\pi
762762 1.94825e12 0.209337
763763 −8.78554e12 −0.938443
764764 −2.55046e12 −0.270831
765765 −1.16923e13 −1.23431
766766 −1.18381e13 −1.24237
767767 −3.92892e12 −0.409915
768768 1.75945e12 0.182495
769769 −5.30538e12 −0.547076 −0.273538 0.961861i 0.588194π-0.588194\pi
−0.273538 + 0.961861i 0.588194π0.588194\pi
770770 −3.21527e13 −3.29616
771771 −3.13349e12 −0.319363
772772 2.19887e12 0.222804
773773 −1.53974e12 −0.155110 −0.0775550 0.996988i 0.524711π-0.524711\pi
−0.0775550 + 0.996988i 0.524711π0.524711\pi
774774 1.29990e12 0.130189
775775 5.80017e12 0.577541
776776 −4.69979e12 −0.465266
777777 −5.26854e12 −0.518556
778778 −9.32896e12 −0.912904
779779 −9.39759e11 −0.0914319
780780 −2.33300e11 −0.0225678
781781 1.19084e13 1.14531
782782 2.12803e13 2.03492
783783 1.19745e12 0.113849
784784 −1.06437e13 −1.00617
785785 −9.18125e12 −0.862955
786786 4.35127e12 0.406644
787787 −2.94759e12 −0.273893 −0.136946 0.990578i 0.543729π-0.543729\pi
−0.136946 + 0.990578i 0.543729π0.543729\pi
788788 1.23615e12 0.114209
789789 2.24738e12 0.206457
790790 1.18747e13 1.08467
791791 1.39257e13 1.26480
792792 1.47524e13 1.33229
793793 1.22692e12 0.110176
794794 −1.94266e13 −1.73462
795795 −9.25425e12 −0.821654
796796 −7.71099e11 −0.0680772
797797 −1.83481e13 −1.61075 −0.805376 0.592764i 0.798037π-0.798037\pi
−0.805376 + 0.592764i 0.798037π0.798037\pi
798798 −3.86139e12 −0.337079
799799 9.75873e12 0.847096
800800 2.76632e12 0.238779
801801 8.74291e12 0.750429
802802 7.51127e12 0.641104
803803 −2.79561e12 −0.237278
804804 2.49549e11 0.0210622
805805 3.65600e13 3.06849
806806 3.03099e12 0.252975
807807 −4.32744e12 −0.359170
808808 6.67470e12 0.550910
809809 −9.70701e12 −0.796741 −0.398371 0.917225i 0.630424π-0.630424\pi
−0.398371 + 0.917225i 0.630424π0.630424\pi
810810 −1.09578e13 −0.894415
811811 6.14421e12 0.498738 0.249369 0.968409i 0.419777π-0.419777\pi
0.249369 + 0.968409i 0.419777π0.419777\pi
812812 5.09203e11 0.0411045
813813 4.18005e12 0.335563
814814 2.46220e13 1.96568
815815 −1.42192e13 −1.12893
816816 −5.62572e12 −0.444194
817817 −1.11457e12 −0.0875205
818818 −2.22030e13 −1.73389
819819 −4.27330e12 −0.331884
820820 4.24720e11 0.0328050
821821 1.41092e13 1.08383 0.541913 0.840434i 0.317700π-0.317700\pi
0.541913 + 0.840434i 0.317700π0.317700\pi
822822 4.66057e12 0.356054
823823 7.99374e12 0.607366 0.303683 0.952773i 0.401784π-0.401784\pi
0.303683 + 0.952773i 0.401784π0.401784\pi
824824 −1.18888e13 −0.898389
825825 −5.55951e12 −0.417824
826826 2.93892e13 2.19674
827827 −4.82994e12 −0.359060 −0.179530 0.983752i 0.557458π-0.557458\pi
−0.179530 + 0.983752i 0.557458π0.557458\pi
828828 3.57834e12 0.264573
829829 1.41648e13 1.04163 0.520816 0.853669i 0.325628π-0.325628\pi
0.520816 + 0.853669i 0.325628π0.325628\pi
830830 4.40666e12 0.322298
831831 8.65637e12 0.629697
832832 −2.94327e12 −0.212949
833833 1.32956e13 0.956762
834834 −4.68031e12 −0.334987
835835 −2.61443e11 −0.0186118
836836 2.69831e12 0.191058
837837 −7.97228e12 −0.561459
838838 −8.24927e12 −0.577853
839839 1.79477e13 1.25049 0.625246 0.780428i 0.284999π-0.284999\pi
0.625246 + 0.780428i 0.284999π0.284999\pi
840840 −8.18088e12 −0.566948
841841 −1.40851e13 −0.970907
842842 4.59068e12 0.314755
843843 3.78778e12 0.258322
844844 2.43291e11 0.0165038
845845 1.48053e12 0.0998993
846846 1.09744e13 0.736570
847847 −3.93372e13 −2.62620
848848 3.06079e13 2.03260
849849 8.44458e12 0.557819
850850 −1.23356e13 −0.810542
851851 −2.79970e13 −1.82990
852852 −6.46347e11 −0.0420231
853853 −8.65463e12 −0.559729 −0.279865 0.960039i 0.590290π-0.590290\pi
−0.279865 + 0.960039i 0.590290π0.590290\pi
854854 −9.17764e12 −0.590433
855855 1.12742e13 0.721505
856856 6.16360e12 0.392376
857857 −1.17199e13 −0.742185 −0.371092 0.928596i 0.621017π-0.621017\pi
−0.371092 + 0.928596i 0.621017π0.621017\pi
858858 −2.90524e12 −0.183016
859859 −2.70022e13 −1.69211 −0.846057 0.533092i 0.821030π-0.821030\pi
−0.846057 + 0.533092i 0.821030π0.821030\pi
860860 5.03727e11 0.0314016
861861 −1.13171e12 −0.0701815
862862 −2.35669e13 −1.45385
863863 2.11111e13 1.29557 0.647786 0.761823i 0.275695π-0.275695\pi
0.647786 + 0.761823i 0.275695π0.275695\pi
864864 −3.80228e12 −0.232130
865865 −2.47060e13 −1.50048
866866 −2.81704e13 −1.70201
867867 1.09830e12 0.0660137
868868 −3.39013e12 −0.202711
869869 2.21107e13 1.31527
870870 1.44645e12 0.0855986
871871 −1.58365e12 −0.0932346
872872 1.04468e13 0.611870
873873 7.79984e12 0.454487
874874 −2.05194e13 −1.18950
875875 9.67354e12 0.557891
876876 1.51737e11 0.00870606
877877 −1.48840e13 −0.849614 −0.424807 0.905284i 0.639658π-0.639658\pi
−0.424807 + 0.905284i 0.639658π0.639658\pi
878878 −1.33544e13 −0.758402
879879 −5.94176e12 −0.335711
880880 4.51687e13 2.53902
881881 8.21496e11 0.0459424 0.0229712 0.999736i 0.492687π-0.492687\pi
0.0229712 + 0.999736i 0.492687π0.492687\pi
882882 1.49518e13 0.831926
883883 −5.06069e12 −0.280147 −0.140074 0.990141i 0.544734π-0.544734\pi
−0.140074 + 0.990141i 0.544734π0.544734\pi
884884 −9.63878e11 −0.0530868
885885 1.24829e13 0.684025
886886 2.74964e12 0.149908
887887 −4.00796e12 −0.217404 −0.108702 0.994074i 0.534669π-0.534669\pi
−0.108702 + 0.994074i 0.534669π0.534669\pi
888888 6.26478e12 0.338102
889889 1.38286e13 0.742540
890890 2.26582e13 1.21051
891891 −2.04035e13 −1.08456
892892 −3.22233e12 −0.170423
893893 −9.40979e12 −0.495163
894894 −1.42484e13 −0.746016
895895 9.04211e11 0.0471049
896896 3.12129e13 1.61788
897897 3.30347e12 0.170374
898898 2.74516e13 1.40872
899899 −2.80989e12 −0.143473
900900 −2.07427e12 −0.105384
901901 −3.82338e13 −1.93280
902902 5.28894e12 0.266035
903903 −1.34224e12 −0.0671791
904904 −1.65589e13 −0.824657
905905 5.20459e12 0.257910
906906 −3.91569e12 −0.193077
907907 4.52189e12 0.221864 0.110932 0.993828i 0.464616π-0.464616\pi
0.110932 + 0.993828i 0.464616π0.464616\pi
908908 −6.79966e12 −0.331971
909909 −1.10774e13 −0.538147
910910 −1.10747e13 −0.535361
911911 −5.68437e12 −0.273432 −0.136716 0.990610i 0.543655π-0.543655\pi
−0.136716 + 0.990610i 0.543655π0.543655\pi
912912 5.42456e12 0.259650
913913 8.20525e12 0.390817
914914 3.40839e13 1.61544
915915 −3.89816e12 −0.183850
916916 −5.12257e12 −0.240413
917917 3.08851e13 1.44241
918918 1.69552e13 0.787972
919919 3.28047e13 1.51711 0.758555 0.651609i 0.225906π-0.225906\pi
0.758555 + 0.651609i 0.225906π0.225906\pi
920920 −4.34731e13 −2.00067
921921 −9.97597e12 −0.456864
922922 −2.60183e13 −1.18574
923923 4.10174e12 0.186020
924924 3.24947e12 0.146652
925925 1.62291e13 0.728880
926926 −6.90877e12 −0.308782
927927 1.97308e13 0.877576
928928 −1.34014e12 −0.0593177
929929 1.82868e13 0.805504 0.402752 0.915309i 0.368054π-0.368054\pi
0.402752 + 0.915309i 0.368054π0.368054\pi
930930 −9.63005e12 −0.422139
931931 −1.28201e13 −0.559267
932932 −4.21280e12 −0.182894
933933 9.50675e12 0.410738
934934 2.43183e13 1.04562
935935 −5.64224e13 −2.41435
936936 5.08134e12 0.216390
937937 2.71335e13 1.14995 0.574974 0.818171i 0.305012π-0.305012\pi
0.574974 + 0.818171i 0.305012π0.305012\pi
938938 1.18461e13 0.499644
939939 −1.69828e12 −0.0712875
940940 4.25271e12 0.177660
941941 −9.48833e12 −0.394490 −0.197245 0.980354i 0.563200π-0.563200\pi
−0.197245 + 0.980354i 0.563200π0.563200\pi
942942 6.20556e12 0.256775
943943 −6.01392e12 −0.247660
944944 −4.12866e13 −1.69213
945945 2.91293e13 1.18819
946946 6.27279e12 0.254654
947947 1.99262e13 0.805100 0.402550 0.915398i 0.368124π-0.368124\pi
0.402550 + 0.915398i 0.368124π0.368124\pi
948948 −1.20010e12 −0.0482591
949949 −9.62925e11 −0.0385385
950950 1.18945e13 0.473796
951951 −4.26759e11 −0.0169188
952952 −3.37992e13 −1.33364
953953 2.48922e13 0.977565 0.488782 0.872406i 0.337441π-0.337441\pi
0.488782 + 0.872406i 0.337441π0.337441\pi
954954 −4.29967e13 −1.68061
955955 −5.14238e13 −2.00055
956956 −7.92144e11 −0.0306721
957957 2.69330e12 0.103796
958958 3.38780e13 1.29949
959959 3.30805e13 1.26296
960960 9.35134e12 0.355347
961961 −7.73217e12 −0.292446
962962 8.48083e12 0.319264
963963 −1.02292e13 −0.383286
964964 −1.58481e12 −0.0591060
965965 4.43349e13 1.64579
966966 −2.47107e13 −0.913037
967967 −4.20642e13 −1.54701 −0.773505 0.633790i 0.781499π-0.781499\pi
−0.773505 + 0.633790i 0.781499π0.781499\pi
968968 4.67756e13 1.71230
969969 −6.77609e12 −0.246900
970970 2.02141e13 0.733132
971971 −1.34740e12 −0.0486418 −0.0243209 0.999704i 0.507742π-0.507742\pi
−0.0243209 + 0.999704i 0.507742π0.507742\pi
972972 4.37325e12 0.157147
973973 −3.32207e13 −1.18823
974974 −2.36825e13 −0.843164
975975 −1.91493e12 −0.0678628
976976 1.28929e13 0.454807
977977 −2.05822e13 −0.722714 −0.361357 0.932428i 0.617686π-0.617686\pi
−0.361357 + 0.932428i 0.617686π0.617686\pi
978978 9.61070e12 0.335916
979979 4.21898e13 1.46786
980980 5.79401e12 0.200661
981981 −1.73377e13 −0.597695
982982 2.71700e13 0.932368
983983 −5.72322e13 −1.95501 −0.977506 0.210906i 0.932359π-0.932359\pi
−0.977506 + 0.210906i 0.932359π0.932359\pi
984984 1.34571e12 0.0457587
985985 2.49239e13 0.843631
986986 5.97598e12 0.201355
987987 −1.13318e13 −0.380078
988988 9.29412e11 0.0310315
989989 −7.13263e12 −0.237065
990990 −6.34511e13 −2.09933
991991 1.46766e13 0.483386 0.241693 0.970353i 0.422297π-0.422297\pi
0.241693 + 0.970353i 0.422297π0.422297\pi
992992 8.92228e12 0.292532
993993 2.54750e12 0.0831462
994994 −3.06820e13 −0.996884
995995 −1.55473e13 −0.502866
996996 −4.45354e11 −0.0143396
997997 −5.31286e13 −1.70294 −0.851471 0.524402i 0.824289π-0.824289\pi
−0.851471 + 0.524402i 0.824289π0.824289\pi
998998 5.51037e13 1.75830
999999 −2.23067e13 −0.708583
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 13.10.a.a.1.2 4
3.2 odd 2 117.10.a.c.1.3 4
4.3 odd 2 208.10.a.g.1.1 4
5.4 even 2 325.10.a.a.1.3 4
13.12 even 2 169.10.a.a.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.10.a.a.1.2 4 1.1 even 1 trivial
117.10.a.c.1.3 4 3.2 odd 2
169.10.a.a.1.3 4 13.12 even 2
208.10.a.g.1.1 4 4.3 odd 2
325.10.a.a.1.3 4 5.4 even 2