Properties

Label 686.2.a.b.1.1
Level 686686
Weight 22
Character 686.1
Self dual yes
Analytic conductor 5.4785.478
Analytic rank 00
Dimension 33
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] = [686,2,Mod(1,686)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(686, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("686.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 686=273 686 = 2 \cdot 7^{3}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 686.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: 5.477737578665.47773757866
Analytic rank: 00
Dimension: 33
Coefficient field: Q(ζ14)+\Q(\zeta_{14})^+
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x3x22x+1 x^{3} - x^{2} - 2x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

Embedding invariants

Embedding label 1.1
Root 1.801941.80194 of defining polynomial
Character χ\chi == 686.1

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) == q1.00000q21.24698q3+1.00000q4+1.35690q5+1.24698q61.00000q81.44504q91.35690q100.335126q111.24698q12+6.38404q131.69202q15+1.00000q162.96077q17+1.44504q18+4.10992q19+1.35690q20+0.335126q223.65279q23+1.24698q243.15883q256.38404q26+5.54288q27+4.04892q29+1.69202q30+0.704103q311.00000q32+0.417895q33+2.96077q341.44504q36+6.74094q374.10992q387.96077q391.35690q40+6.60388q414.91185q430.335126q441.96077q45+3.65279q460.862937q471.24698q48+3.15883q50+3.69202q51+6.38404q52+12.8877q535.54288q540.454731q555.12498q574.04892q58+9.05861q591.69202q60+13.1588q610.704103q62+1.00000q64+8.66248q650.417895q66+3.14914q672.96077q68+4.55496q69+10.5700q71+1.44504q72+10.0218q736.74094q74+3.93900q75+4.10992q76+7.96077q788.86294q79+1.35690q802.57673q816.60388q8217.4426q834.01746q85+4.91185q865.04892q87+0.335126q880.625646q89+1.96077q903.65279q920.878002q93+0.862937q94+5.57673q95+1.24698q96+3.24698q97+0.484271q99+O(q100)q-1.00000 q^{2} -1.24698 q^{3} +1.00000 q^{4} +1.35690 q^{5} +1.24698 q^{6} -1.00000 q^{8} -1.44504 q^{9} -1.35690 q^{10} -0.335126 q^{11} -1.24698 q^{12} +6.38404 q^{13} -1.69202 q^{15} +1.00000 q^{16} -2.96077 q^{17} +1.44504 q^{18} +4.10992 q^{19} +1.35690 q^{20} +0.335126 q^{22} -3.65279 q^{23} +1.24698 q^{24} -3.15883 q^{25} -6.38404 q^{26} +5.54288 q^{27} +4.04892 q^{29} +1.69202 q^{30} +0.704103 q^{31} -1.00000 q^{32} +0.417895 q^{33} +2.96077 q^{34} -1.44504 q^{36} +6.74094 q^{37} -4.10992 q^{38} -7.96077 q^{39} -1.35690 q^{40} +6.60388 q^{41} -4.91185 q^{43} -0.335126 q^{44} -1.96077 q^{45} +3.65279 q^{46} -0.862937 q^{47} -1.24698 q^{48} +3.15883 q^{50} +3.69202 q^{51} +6.38404 q^{52} +12.8877 q^{53} -5.54288 q^{54} -0.454731 q^{55} -5.12498 q^{57} -4.04892 q^{58} +9.05861 q^{59} -1.69202 q^{60} +13.1588 q^{61} -0.704103 q^{62} +1.00000 q^{64} +8.66248 q^{65} -0.417895 q^{66} +3.14914 q^{67} -2.96077 q^{68} +4.55496 q^{69} +10.5700 q^{71} +1.44504 q^{72} +10.0218 q^{73} -6.74094 q^{74} +3.93900 q^{75} +4.10992 q^{76} +7.96077 q^{78} -8.86294 q^{79} +1.35690 q^{80} -2.57673 q^{81} -6.60388 q^{82} -17.4426 q^{83} -4.01746 q^{85} +4.91185 q^{86} -5.04892 q^{87} +0.335126 q^{88} -0.625646 q^{89} +1.96077 q^{90} -3.65279 q^{92} -0.878002 q^{93} +0.862937 q^{94} +5.57673 q^{95} +1.24698 q^{96} +3.24698 q^{97} +0.484271 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 3q3q2+q3+3q4q63q84q9+q12+9q13+3q16+4q17+4q18+13q19+7q23q24q259q262q27+3q29+16q31++14q99+O(q100) 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 3 q^{8} - 4 q^{9} + q^{12} + 9 q^{13} + 3 q^{16} + 4 q^{17} + 4 q^{18} + 13 q^{19} + 7 q^{23} - q^{24} - q^{25} - 9 q^{26} - 2 q^{27} + 3 q^{29} + 16 q^{31}+ \cdots + 14 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 −1.00000 −0.707107
33 −1.24698 −0.719944 −0.359972 0.932963i 0.617214π-0.617214\pi
−0.359972 + 0.932963i 0.617214π0.617214\pi
44 1.00000 0.500000
55 1.35690 0.606822 0.303411 0.952860i 0.401874π-0.401874\pi
0.303411 + 0.952860i 0.401874π0.401874\pi
66 1.24698 0.509077
77 0 0
88 −1.00000 −0.353553
99 −1.44504 −0.481681
1010 −1.35690 −0.429088
1111 −0.335126 −0.101044 −0.0505221 0.998723i 0.516089π-0.516089\pi
−0.0505221 + 0.998723i 0.516089π0.516089\pi
1212 −1.24698 −0.359972
1313 6.38404 1.77061 0.885307 0.465006i 0.153948π-0.153948\pi
0.885307 + 0.465006i 0.153948π0.153948\pi
1414 0 0
1515 −1.69202 −0.436878
1616 1.00000 0.250000
1717 −2.96077 −0.718093 −0.359046 0.933320i 0.616898π-0.616898\pi
−0.359046 + 0.933320i 0.616898π0.616898\pi
1818 1.44504 0.340600
1919 4.10992 0.942879 0.471440 0.881898i 0.343734π-0.343734\pi
0.471440 + 0.881898i 0.343734π0.343734\pi
2020 1.35690 0.303411
2121 0 0
2222 0.335126 0.0714490
2323 −3.65279 −0.761660 −0.380830 0.924645i 0.624362π-0.624362\pi
−0.380830 + 0.924645i 0.624362π0.624362\pi
2424 1.24698 0.254539
2525 −3.15883 −0.631767
2626 −6.38404 −1.25201
2727 5.54288 1.06673
2828 0 0
2929 4.04892 0.751865 0.375933 0.926647i 0.377322π-0.377322\pi
0.375933 + 0.926647i 0.377322π0.377322\pi
3030 1.69202 0.308919
3131 0.704103 0.126461 0.0632303 0.997999i 0.479860π-0.479860\pi
0.0632303 + 0.997999i 0.479860π0.479860\pi
3232 −1.00000 −0.176777
3333 0.417895 0.0727461
3434 2.96077 0.507768
3535 0 0
3636 −1.44504 −0.240840
3737 6.74094 1.10820 0.554102 0.832449i 0.313062π-0.313062\pi
0.554102 + 0.832449i 0.313062π0.313062\pi
3838 −4.10992 −0.666716
3939 −7.96077 −1.27474
4040 −1.35690 −0.214544
4141 6.60388 1.03135 0.515676 0.856784i 0.327541π-0.327541\pi
0.515676 + 0.856784i 0.327541π0.327541\pi
4242 0 0
4343 −4.91185 −0.749051 −0.374525 0.927217i 0.622194π-0.622194\pi
−0.374525 + 0.927217i 0.622194π0.622194\pi
4444 −0.335126 −0.0505221
4545 −1.96077 −0.292295
4646 3.65279 0.538575
4747 −0.862937 −0.125872 −0.0629361 0.998018i 0.520046π-0.520046\pi
−0.0629361 + 0.998018i 0.520046π0.520046\pi
4848 −1.24698 −0.179986
4949 0 0
5050 3.15883 0.446727
5151 3.69202 0.516986
5252 6.38404 0.885307
5353 12.8877 1.77026 0.885130 0.465343i 0.154069π-0.154069\pi
0.885130 + 0.465343i 0.154069π0.154069\pi
5454 −5.54288 −0.754290
5555 −0.454731 −0.0613159
5656 0 0
5757 −5.12498 −0.678820
5858 −4.04892 −0.531649
5959 9.05861 1.17933 0.589665 0.807648i 0.299260π-0.299260\pi
0.589665 + 0.807648i 0.299260π0.299260\pi
6060 −1.69202 −0.218439
6161 13.1588 1.68482 0.842408 0.538840i 0.181137π-0.181137\pi
0.842408 + 0.538840i 0.181137π0.181137\pi
6262 −0.704103 −0.0894212
6363 0 0
6464 1.00000 0.125000
6565 8.66248 1.07445
6666 −0.417895 −0.0514393
6767 3.14914 0.384729 0.192365 0.981324i 0.438384π-0.438384\pi
0.192365 + 0.981324i 0.438384π0.438384\pi
6868 −2.96077 −0.359046
6969 4.55496 0.548353
7070 0 0
7171 10.5700 1.25443 0.627216 0.778846i 0.284195π-0.284195\pi
0.627216 + 0.778846i 0.284195π0.284195\pi
7272 1.44504 0.170300
7373 10.0218 1.17296 0.586480 0.809964i 0.300513π-0.300513\pi
0.586480 + 0.809964i 0.300513π0.300513\pi
7474 −6.74094 −0.783618
7575 3.93900 0.454837
7676 4.10992 0.471440
7777 0 0
7878 7.96077 0.901380
7979 −8.86294 −0.997158 −0.498579 0.866844i 0.666145π-0.666145\pi
−0.498579 + 0.866844i 0.666145π0.666145\pi
8080 1.35690 0.151706
8181 −2.57673 −0.286303
8282 −6.60388 −0.729276
8383 −17.4426 −1.91458 −0.957290 0.289130i 0.906634π-0.906634\pi
−0.957290 + 0.289130i 0.906634π0.906634\pi
8484 0 0
8585 −4.01746 −0.435755
8686 4.91185 0.529659
8787 −5.04892 −0.541301
8888 0.335126 0.0357245
8989 −0.625646 −0.0663183 −0.0331592 0.999450i 0.510557π-0.510557\pi
−0.0331592 + 0.999450i 0.510557π0.510557\pi
9090 1.96077 0.206683
9191 0 0
9292 −3.65279 −0.380830
9393 −0.878002 −0.0910446
9494 0.862937 0.0890051
9595 5.57673 0.572160
9696 1.24698 0.127269
9797 3.24698 0.329681 0.164840 0.986320i 0.447289π-0.447289\pi
0.164840 + 0.986320i 0.447289π0.447289\pi
9898 0 0
9999 0.484271 0.0486710
100100 −3.15883 −0.315883
101101 1.77479 0.176598 0.0882991 0.996094i 0.471857π-0.471857\pi
0.0882991 + 0.996094i 0.471857π0.471857\pi
102102 −3.69202 −0.365565
103103 15.7168 1.54862 0.774310 0.632807i 0.218097π-0.218097\pi
0.774310 + 0.632807i 0.218097π0.218097\pi
104104 −6.38404 −0.626007
105105 0 0
106106 −12.8877 −1.25176
107107 −17.4306 −1.68508 −0.842538 0.538636i 0.818940π-0.818940\pi
−0.842538 + 0.538636i 0.818940π0.818940\pi
108108 5.54288 0.533364
109109 7.09246 0.679334 0.339667 0.940546i 0.389686π-0.389686\pi
0.339667 + 0.940546i 0.389686π0.389686\pi
110110 0.454731 0.0433569
111111 −8.40581 −0.797844
112112 0 0
113113 −12.9433 −1.21760 −0.608802 0.793322i 0.708350π-0.708350\pi
−0.608802 + 0.793322i 0.708350π0.708350\pi
114114 5.12498 0.479999
115115 −4.95646 −0.462192
116116 4.04892 0.375933
117117 −9.22521 −0.852871
118118 −9.05861 −0.833912
119119 0 0
120120 1.69202 0.154460
121121 −10.8877 −0.989790
122122 −13.1588 −1.19134
123123 −8.23490 −0.742516
124124 0.704103 0.0632303
125125 −11.0707 −0.990192
126126 0 0
127127 10.9487 0.971539 0.485770 0.874087i 0.338539π-0.338539\pi
0.485770 + 0.874087i 0.338539π0.338539\pi
128128 −1.00000 −0.0883883
129129 6.12498 0.539275
130130 −8.66248 −0.759750
131131 −4.43967 −0.387895 −0.193948 0.981012i 0.562129π-0.562129\pi
−0.193948 + 0.981012i 0.562129π0.562129\pi
132132 0.417895 0.0363731
133133 0 0
134134 −3.14914 −0.272045
135135 7.52111 0.647314
136136 2.96077 0.253884
137137 −15.7627 −1.34670 −0.673350 0.739324i 0.735145π-0.735145\pi
−0.673350 + 0.739324i 0.735145π0.735145\pi
138138 −4.55496 −0.387744
139139 −17.9584 −1.52321 −0.761605 0.648042i 0.775588π-0.775588\pi
−0.761605 + 0.648042i 0.775588π0.775588\pi
140140 0 0
141141 1.07606 0.0906210
142142 −10.5700 −0.887017
143143 −2.13946 −0.178910
144144 −1.44504 −0.120420
145145 5.49396 0.456248
146146 −10.0218 −0.829408
147147 0 0
148148 6.74094 0.554102
149149 14.0911 1.15439 0.577195 0.816606i 0.304147π-0.304147\pi
0.577195 + 0.816606i 0.304147π0.304147\pi
150150 −3.93900 −0.321618
151151 12.2567 0.997434 0.498717 0.866765i 0.333805π-0.333805\pi
0.498717 + 0.866765i 0.333805π0.333805\pi
152152 −4.10992 −0.333358
153153 4.27844 0.345891
154154 0 0
155155 0.955395 0.0767391
156156 −7.96077 −0.637372
157157 −21.2054 −1.69237 −0.846186 0.532888i 0.821107π-0.821107\pi
−0.846186 + 0.532888i 0.821107π0.821107\pi
158158 8.86294 0.705097
159159 −16.0707 −1.27449
160160 −1.35690 −0.107272
161161 0 0
162162 2.57673 0.202447
163163 18.4862 1.44795 0.723975 0.689826i 0.242313π-0.242313\pi
0.723975 + 0.689826i 0.242313π0.242313\pi
164164 6.60388 0.515676
165165 0.567040 0.0441440
166166 17.4426 1.35381
167167 −20.2741 −1.56886 −0.784430 0.620218i 0.787044π-0.787044\pi
−0.784430 + 0.620218i 0.787044π0.787044\pi
168168 0 0
169169 27.7560 2.13508
170170 4.01746 0.308125
171171 −5.93900 −0.454167
172172 −4.91185 −0.374525
173173 −6.35690 −0.483306 −0.241653 0.970363i 0.577690π-0.577690\pi
−0.241653 + 0.970363i 0.577690π0.577690\pi
174174 5.04892 0.382757
175175 0 0
176176 −0.335126 −0.0252610
177177 −11.2959 −0.849052
178178 0.625646 0.0468941
179179 3.46011 0.258621 0.129310 0.991604i 0.458724π-0.458724\pi
0.129310 + 0.991604i 0.458724π0.458724\pi
180180 −1.96077 −0.146147
181181 13.7071 1.01884 0.509420 0.860518i 0.329860π-0.329860\pi
0.509420 + 0.860518i 0.329860π0.329860\pi
182182 0 0
183183 −16.4088 −1.21297
184184 3.65279 0.269287
185185 9.14675 0.672483
186186 0.878002 0.0643782
187187 0.992230 0.0725591
188188 −0.862937 −0.0629361
189189 0 0
190190 −5.57673 −0.404578
191191 −6.92931 −0.501387 −0.250694 0.968066i 0.580659π-0.580659\pi
−0.250694 + 0.968066i 0.580659π0.580659\pi
192192 −1.24698 −0.0899930
193193 −0.716185 −0.0515521 −0.0257760 0.999668i 0.508206π-0.508206\pi
−0.0257760 + 0.999668i 0.508206π0.508206\pi
194194 −3.24698 −0.233120
195195 −10.8019 −0.773543
196196 0 0
197197 2.06100 0.146840 0.0734200 0.997301i 0.476609π-0.476609\pi
0.0734200 + 0.997301i 0.476609π0.476609\pi
198198 −0.484271 −0.0344156
199199 0.0217703 0.00154325 0.000771627 1.00000i 0.499754π-0.499754\pi
0.000771627 1.00000i 0.499754π0.499754\pi
200200 3.15883 0.223363
201201 −3.92692 −0.276984
202202 −1.77479 −0.124874
203203 0 0
204204 3.69202 0.258493
205205 8.96077 0.625847
206206 −15.7168 −1.09504
207207 5.27844 0.366877
208208 6.38404 0.442654
209209 −1.37734 −0.0952725
210210 0 0
211211 −27.7875 −1.91297 −0.956484 0.291785i 0.905751π-0.905751\pi
−0.956484 + 0.291785i 0.905751π0.905751\pi
212212 12.8877 0.885130
213213 −13.1806 −0.903120
214214 17.4306 1.19153
215215 −6.66487 −0.454541
216216 −5.54288 −0.377145
217217 0 0
218218 −7.09246 −0.480362
219219 −12.4969 −0.844465
220220 −0.454731 −0.0306579
221221 −18.9017 −1.27147
222222 8.40581 0.564161
223223 −12.7017 −0.850569 −0.425285 0.905060i 0.639826π-0.639826\pi
−0.425285 + 0.905060i 0.639826π0.639826\pi
224224 0 0
225225 4.56465 0.304310
226226 12.9433 0.860976
227227 −6.49635 −0.431178 −0.215589 0.976484i 0.569167π-0.569167\pi
−0.215589 + 0.976484i 0.569167π0.569167\pi
228228 −5.12498 −0.339410
229229 9.70171 0.641107 0.320554 0.947230i 0.396131π-0.396131\pi
0.320554 + 0.947230i 0.396131π0.396131\pi
230230 4.95646 0.326819
231231 0 0
232232 −4.04892 −0.265824
233233 1.81940 0.119193 0.0595963 0.998223i 0.481019π-0.481019\pi
0.0595963 + 0.998223i 0.481019π0.481019\pi
234234 9.22521 0.603071
235235 −1.17092 −0.0763821
236236 9.05861 0.589665
237237 11.0519 0.717898
238238 0 0
239239 8.19567 0.530134 0.265067 0.964230i 0.414606π-0.414606\pi
0.265067 + 0.964230i 0.414606π0.414606\pi
240240 −1.69202 −0.109220
241241 5.23729 0.337364 0.168682 0.985671i 0.446049π-0.446049\pi
0.168682 + 0.985671i 0.446049π0.446049\pi
242242 10.8877 0.699887
243243 −13.4155 −0.860605
244244 13.1588 0.842408
245245 0 0
246246 8.23490 0.525038
247247 26.2379 1.66948
248248 −0.704103 −0.0447106
249249 21.7506 1.37839
250250 11.0707 0.700172
251251 0.972853 0.0614059 0.0307030 0.999529i 0.490225π-0.490225\pi
0.0307030 + 0.999529i 0.490225π0.490225\pi
252252 0 0
253253 1.22414 0.0769613
254254 −10.9487 −0.686982
255255 5.00969 0.313719
256256 1.00000 0.0625000
257257 −15.7168 −0.980386 −0.490193 0.871614i 0.663074π-0.663074\pi
−0.490193 + 0.871614i 0.663074π0.663074\pi
258258 −6.12498 −0.381325
259259 0 0
260260 8.66248 0.537224
261261 −5.85086 −0.362159
262262 4.43967 0.274283
263263 −18.5623 −1.14460 −0.572299 0.820045i 0.693948π-0.693948\pi
−0.572299 + 0.820045i 0.693948π0.693948\pi
264264 −0.417895 −0.0257196
265265 17.4873 1.07423
266266 0 0
267267 0.780167 0.0477455
268268 3.14914 0.192365
269269 11.1304 0.678630 0.339315 0.940673i 0.389805π-0.389805\pi
0.339315 + 0.940673i 0.389805π0.389805\pi
270270 −7.52111 −0.457720
271271 20.1250 1.22251 0.611253 0.791435i 0.290666π-0.290666\pi
0.611253 + 0.791435i 0.290666π0.290666\pi
272272 −2.96077 −0.179523
273273 0 0
274274 15.7627 0.952260
275275 1.05861 0.0638363
276276 4.55496 0.274176
277277 −25.8485 −1.55308 −0.776542 0.630066i 0.783028π-0.783028\pi
−0.776542 + 0.630066i 0.783028π0.783028\pi
278278 17.9584 1.07707
279279 −1.01746 −0.0609136
280280 0 0
281281 −13.0737 −0.779910 −0.389955 0.920834i 0.627509π-0.627509\pi
−0.389955 + 0.920834i 0.627509π0.627509\pi
282282 −1.07606 −0.0640787
283283 5.77718 0.343418 0.171709 0.985148i 0.445071π-0.445071\pi
0.171709 + 0.985148i 0.445071π0.445071\pi
284284 10.5700 0.627216
285285 −6.95407 −0.411923
286286 2.13946 0.126509
287287 0 0
288288 1.44504 0.0851499
289289 −8.23383 −0.484343
290290 −5.49396 −0.322616
291291 −4.04892 −0.237352
292292 10.0218 0.586480
293293 20.5200 1.19879 0.599397 0.800452i 0.295407π-0.295407\pi
0.599397 + 0.800452i 0.295407π0.295407\pi
294294 0 0
295295 12.2916 0.715644
296296 −6.74094 −0.391809
297297 −1.85756 −0.107787
298298 −14.0911 −0.816277
299299 −23.3196 −1.34861
300300 3.93900 0.227418
301301 0 0
302302 −12.2567 −0.705292
303303 −2.21313 −0.127141
304304 4.10992 0.235720
305305 17.8552 1.02238
306306 −4.27844 −0.244582
307307 10.2446 0.584689 0.292345 0.956313i 0.405565π-0.405565\pi
0.292345 + 0.956313i 0.405565π0.405565\pi
308308 0 0
309309 −19.5985 −1.11492
310310 −0.955395 −0.0542628
311311 34.6679 1.96583 0.982917 0.184050i 0.0589207π-0.0589207\pi
0.982917 + 0.184050i 0.0589207π0.0589207\pi
312312 7.96077 0.450690
313313 −11.6993 −0.661285 −0.330642 0.943756i 0.607265π-0.607265\pi
−0.330642 + 0.943756i 0.607265π0.607265\pi
314314 21.2054 1.19669
315315 0 0
316316 −8.86294 −0.498579
317317 −6.82908 −0.383560 −0.191780 0.981438i 0.561426π-0.561426\pi
−0.191780 + 0.981438i 0.561426π0.561426\pi
318318 16.0707 0.901199
319319 −1.35690 −0.0759716
320320 1.35690 0.0758528
321321 21.7356 1.21316
322322 0 0
323323 −12.1685 −0.677075
324324 −2.57673 −0.143152
325325 −20.1661 −1.11862
326326 −18.4862 −1.02386
327327 −8.84415 −0.489083
328328 −6.60388 −0.364638
329329 0 0
330330 −0.567040 −0.0312145
331331 −15.2664 −0.839115 −0.419557 0.907729i 0.637815π-0.637815\pi
−0.419557 + 0.907729i 0.637815π0.637815\pi
332332 −17.4426 −0.957290
333333 −9.74094 −0.533800
334334 20.2741 1.10935
335335 4.27306 0.233462
336336 0 0
337337 −0.0163935 −0.000893008 0 −0.000446504 1.00000i 0.500142π-0.500142\pi
−0.000446504 1.00000i 0.500142π0.500142\pi
338338 −27.7560 −1.50973
339339 16.1400 0.876607
340340 −4.01746 −0.217877
341341 −0.235963 −0.0127781
342342 5.93900 0.321144
343343 0 0
344344 4.91185 0.264829
345345 6.18060 0.332753
346346 6.35690 0.341749
347347 19.5133 1.04753 0.523765 0.851863i 0.324527π-0.324527\pi
0.523765 + 0.851863i 0.324527π0.324527\pi
348348 −5.04892 −0.270650
349349 11.7778 0.630450 0.315225 0.949017i 0.397920π-0.397920\pi
0.315225 + 0.949017i 0.397920π0.397920\pi
350350 0 0
351351 35.3860 1.88876
352352 0.335126 0.0178623
353353 34.1672 1.81854 0.909268 0.416211i 0.136642π-0.136642\pi
0.909268 + 0.416211i 0.136642π0.136642\pi
354354 11.2959 0.600370
355355 14.3424 0.761217
356356 −0.625646 −0.0331592
357357 0 0
358358 −3.46011 −0.182872
359359 10.6558 0.562390 0.281195 0.959651i 0.409269π-0.409269\pi
0.281195 + 0.959651i 0.409269π0.409269\pi
360360 1.96077 0.103342
361361 −2.10859 −0.110978
362362 −13.7071 −0.720428
363363 13.5767 0.712593
364364 0 0
365365 13.5985 0.711778
366366 16.4088 0.857702
367367 −33.0978 −1.72769 −0.863846 0.503755i 0.831951π-0.831951\pi
−0.863846 + 0.503755i 0.831951π0.831951\pi
368368 −3.65279 −0.190415
369369 −9.54288 −0.496782
370370 −9.14675 −0.475517
371371 0 0
372372 −0.878002 −0.0455223
373373 −18.5754 −0.961798 −0.480899 0.876776i 0.659690π-0.659690\pi
−0.480899 + 0.876776i 0.659690π0.659690\pi
374374 −0.992230 −0.0513070
375375 13.8049 0.712883
376376 0.862937 0.0445026
377377 25.8485 1.33126
378378 0 0
379379 8.85517 0.454859 0.227430 0.973795i 0.426968π-0.426968\pi
0.227430 + 0.973795i 0.426968π0.426968\pi
380380 5.57673 0.286080
381381 −13.6528 −0.699454
382382 6.92931 0.354534
383383 −30.6969 −1.56854 −0.784270 0.620420i 0.786962π-0.786962\pi
−0.784270 + 0.620420i 0.786962π0.786962\pi
384384 1.24698 0.0636347
385385 0 0
386386 0.716185 0.0364528
387387 7.09783 0.360803
388388 3.24698 0.164840
389389 −1.26205 −0.0639882 −0.0319941 0.999488i 0.510186π-0.510186\pi
−0.0319941 + 0.999488i 0.510186π0.510186\pi
390390 10.8019 0.546977
391391 10.8151 0.546942
392392 0 0
393393 5.53617 0.279263
394394 −2.06100 −0.103832
395395 −12.0261 −0.605098
396396 0.484271 0.0243355
397397 −12.4558 −0.625138 −0.312569 0.949895i 0.601190π-0.601190\pi
−0.312569 + 0.949895i 0.601190π0.601190\pi
398398 −0.0217703 −0.00109124
399399 0 0
400400 −3.15883 −0.157942
401401 −11.8267 −0.590597 −0.295298 0.955405i 0.595419π-0.595419\pi
−0.295298 + 0.955405i 0.595419π0.595419\pi
402402 3.92692 0.195857
403403 4.49502 0.223913
404404 1.77479 0.0882991
405405 −3.49635 −0.173735
406406 0 0
407407 −2.25906 −0.111978
408408 −3.69202 −0.182782
409409 18.4058 0.910109 0.455054 0.890464i 0.349620π-0.349620\pi
0.455054 + 0.890464i 0.349620π0.349620\pi
410410 −8.96077 −0.442541
411411 19.6558 0.969548
412412 15.7168 0.774310
413413 0 0
414414 −5.27844 −0.259421
415415 −23.6679 −1.16181
416416 −6.38404 −0.313003
417417 22.3937 1.09663
418418 1.37734 0.0673678
419419 −5.12067 −0.250161 −0.125081 0.992147i 0.539919π-0.539919\pi
−0.125081 + 0.992147i 0.539919π0.539919\pi
420420 0 0
421421 −16.4034 −0.799454 −0.399727 0.916634i 0.630895π-0.630895\pi
−0.399727 + 0.916634i 0.630895π0.630895\pi
422422 27.7875 1.35267
423423 1.24698 0.0606302
424424 −12.8877 −0.625882
425425 9.35258 0.453667
426426 13.1806 0.638602
427427 0 0
428428 −17.4306 −0.842538
429429 2.66786 0.128805
430430 6.66487 0.321409
431431 −0.330814 −0.0159347 −0.00796737 0.999968i 0.502536π-0.502536\pi
−0.00796737 + 0.999968i 0.502536π0.502536\pi
432432 5.54288 0.266682
433433 −2.45042 −0.117760 −0.0588798 0.998265i 0.518753π-0.518753\pi
−0.0588798 + 0.998265i 0.518753π0.518753\pi
434434 0 0
435435 −6.85086 −0.328473
436436 7.09246 0.339667
437437 −15.0127 −0.718154
438438 12.4969 0.597127
439439 2.69202 0.128483 0.0642416 0.997934i 0.479537π-0.479537\pi
0.0642416 + 0.997934i 0.479537π0.479537\pi
440440 0.454731 0.0216784
441441 0 0
442442 18.9017 0.899062
443443 36.5526 1.73666 0.868332 0.495983i 0.165192π-0.165192\pi
0.868332 + 0.495983i 0.165192π0.165192\pi
444444 −8.40581 −0.398922
445445 −0.848936 −0.0402434
446446 12.7017 0.601443
447447 −17.5714 −0.831096
448448 0 0
449449 −3.48427 −0.164433 −0.0822164 0.996614i 0.526200π-0.526200\pi
−0.0822164 + 0.996614i 0.526200π0.526200\pi
450450 −4.56465 −0.215180
451451 −2.21313 −0.104212
452452 −12.9433 −0.608802
453453 −15.2838 −0.718096
454454 6.49635 0.304889
455455 0 0
456456 5.12498 0.239999
457457 −23.1183 −1.08143 −0.540714 0.841207i 0.681846π-0.681846\pi
−0.540714 + 0.841207i 0.681846π0.681846\pi
458458 −9.70171 −0.453331
459459 −16.4112 −0.766009
460460 −4.95646 −0.231096
461461 9.69501 0.451541 0.225771 0.974180i 0.427510π-0.427510\pi
0.225771 + 0.974180i 0.427510π0.427510\pi
462462 0 0
463463 −5.84654 −0.271712 −0.135856 0.990729i 0.543378π-0.543378\pi
−0.135856 + 0.990729i 0.543378π0.543378\pi
464464 4.04892 0.187966
465465 −1.19136 −0.0552479
466466 −1.81940 −0.0842819
467467 −25.7832 −1.19310 −0.596551 0.802575i 0.703463π-0.703463\pi
−0.596551 + 0.802575i 0.703463π0.703463\pi
468468 −9.22521 −0.426435
469469 0 0
470470 1.17092 0.0540103
471471 26.4426 1.21841
472472 −9.05861 −0.416956
473473 1.64609 0.0756872
474474 −11.0519 −0.507631
475475 −12.9825 −0.595680
476476 0 0
477477 −18.6233 −0.852700
478478 −8.19567 −0.374861
479479 20.8799 0.954028 0.477014 0.878896i 0.341719π-0.341719\pi
0.477014 + 0.878896i 0.341719π0.341719\pi
480480 1.69202 0.0772299
481481 43.0344 1.96220
482482 −5.23729 −0.238552
483483 0 0
484484 −10.8877 −0.494895
485485 4.40581 0.200058
486486 13.4155 0.608540
487487 −0.629958 −0.0285461 −0.0142731 0.999898i 0.504543π-0.504543\pi
−0.0142731 + 0.999898i 0.504543π0.504543\pi
488488 −13.1588 −0.595672
489489 −23.0519 −1.04244
490490 0 0
491491 −7.19269 −0.324601 −0.162301 0.986741i 0.551891π-0.551891\pi
−0.162301 + 0.986741i 0.551891π0.551891\pi
492492 −8.23490 −0.371258
493493 −11.9879 −0.539909
494494 −26.2379 −1.18050
495495 0.657105 0.0295347
496496 0.704103 0.0316152
497497 0 0
498498 −21.7506 −0.974669
499499 21.2567 0.951579 0.475790 0.879559i 0.342162π-0.342162\pi
0.475790 + 0.879559i 0.342162π0.342162\pi
500500 −11.0707 −0.495096
501501 25.2814 1.12949
502502 −0.972853 −0.0434206
503503 25.7614 1.14864 0.574322 0.818630i 0.305266π-0.305266\pi
0.574322 + 0.818630i 0.305266π0.305266\pi
504504 0 0
505505 2.40821 0.107164
506506 −1.22414 −0.0544199
507507 −34.6112 −1.53714
508508 10.9487 0.485770
509509 17.1967 0.762232 0.381116 0.924527i 0.375540π-0.375540\pi
0.381116 + 0.924527i 0.375540π0.375540\pi
510510 −5.00969 −0.221833
511511 0 0
512512 −1.00000 −0.0441942
513513 22.7808 1.00580
514514 15.7168 0.693237
515515 21.3260 0.939737
516516 6.12498 0.269637
517517 0.289192 0.0127187
518518 0 0
519519 7.92692 0.347953
520520 −8.66248 −0.379875
521521 −0.432960 −0.0189683 −0.00948417 0.999955i 0.503019π-0.503019\pi
−0.00948417 + 0.999955i 0.503019π0.503019\pi
522522 5.85086 0.256085
523523 32.9071 1.43893 0.719463 0.694531i 0.244388π-0.244388\pi
0.719463 + 0.694531i 0.244388π0.244388\pi
524524 −4.43967 −0.193948
525525 0 0
526526 18.5623 0.809353
527527 −2.08469 −0.0908104
528528 0.417895 0.0181865
529529 −9.65710 −0.419874
530530 −17.4873 −0.759598
531531 −13.0901 −0.568060
532532 0 0
533533 42.1594 1.82613
534534 −0.780167 −0.0337611
535535 −23.6515 −1.02254
536536 −3.14914 −0.136022
537537 −4.31468 −0.186192
538538 −11.1304 −0.479864
539539 0 0
540540 7.52111 0.323657
541541 10.3056 0.443072 0.221536 0.975152i 0.428893π-0.428893\pi
0.221536 + 0.975152i 0.428893π0.428893\pi
542542 −20.1250 −0.864442
543543 −17.0925 −0.733508
544544 2.96077 0.126942
545545 9.62373 0.412235
546546 0 0
547547 4.55363 0.194699 0.0973496 0.995250i 0.468964π-0.468964\pi
0.0973496 + 0.995250i 0.468964π0.468964\pi
548548 −15.7627 −0.673350
549549 −19.0151 −0.811543
550550 −1.05861 −0.0451391
551551 16.6407 0.708918
552552 −4.55496 −0.193872
553553 0 0
554554 25.8485 1.09820
555555 −11.4058 −0.484150
556556 −17.9584 −0.761605
557557 −28.2911 −1.19873 −0.599366 0.800475i 0.704581π-0.704581\pi
−0.599366 + 0.800475i 0.704581π0.704581\pi
558558 1.01746 0.0430724
559559 −31.3575 −1.32628
560560 0 0
561561 −1.23729 −0.0522385
562562 13.0737 0.551480
563563 26.1062 1.10024 0.550122 0.835084i 0.314581π-0.314581\pi
0.550122 + 0.835084i 0.314581π0.314581\pi
564564 1.07606 0.0453105
565565 −17.5627 −0.738870
566566 −5.77718 −0.242833
567567 0 0
568568 −10.5700 −0.443508
569569 −9.66248 −0.405072 −0.202536 0.979275i 0.564918π-0.564918\pi
−0.202536 + 0.979275i 0.564918π0.564918\pi
570570 6.95407 0.291274
571571 −17.5415 −0.734091 −0.367045 0.930203i 0.619631π-0.619631\pi
−0.367045 + 0.930203i 0.619631π0.619631\pi
572572 −2.13946 −0.0894552
573573 8.64071 0.360971
574574 0 0
575575 11.5386 0.481191
576576 −1.44504 −0.0602101
577577 −5.04593 −0.210065 −0.105032 0.994469i 0.533495π-0.533495\pi
−0.105032 + 0.994469i 0.533495π0.533495\pi
578578 8.23383 0.342482
579579 0.893068 0.0371146
580580 5.49396 0.228124
581581 0 0
582582 4.04892 0.167833
583583 −4.31900 −0.178875
584584 −10.0218 −0.414704
585585 −12.5176 −0.517541
586586 −20.5200 −0.847675
587587 −23.3787 −0.964941 −0.482470 0.875912i 0.660260π-0.660260\pi
−0.482470 + 0.875912i 0.660260π0.660260\pi
588588 0 0
589589 2.89380 0.119237
590590 −12.2916 −0.506037
591591 −2.57002 −0.105717
592592 6.74094 0.277051
593593 21.4330 0.880146 0.440073 0.897962i 0.354953π-0.354953\pi
0.440073 + 0.897962i 0.354953π0.354953\pi
594594 1.85756 0.0762166
595595 0 0
596596 14.0911 0.577195
597597 −0.0271471 −0.00111106
598598 23.3196 0.953609
599599 −33.8170 −1.38173 −0.690863 0.722986i 0.742769π-0.742769\pi
−0.690863 + 0.722986i 0.742769π0.742769\pi
600600 −3.93900 −0.160809
601601 −44.4161 −1.81177 −0.905885 0.423524i 0.860793π-0.860793\pi
−0.905885 + 0.423524i 0.860793π0.860793\pi
602602 0 0
603603 −4.55065 −0.185317
604604 12.2567 0.498717
605605 −14.7735 −0.600627
606606 2.21313 0.0899022
607607 5.74525 0.233193 0.116596 0.993179i 0.462802π-0.462802\pi
0.116596 + 0.993179i 0.462802π0.462802\pi
608608 −4.10992 −0.166679
609609 0 0
610610 −17.8552 −0.722935
611611 −5.50902 −0.222871
612612 4.27844 0.172946
613613 40.4741 1.63473 0.817367 0.576117i 0.195433π-0.195433\pi
0.817367 + 0.576117i 0.195433π0.195433\pi
614614 −10.2446 −0.413438
615615 −11.1739 −0.450575
616616 0 0
617617 −35.3706 −1.42397 −0.711984 0.702196i 0.752203π-0.752203\pi
−0.711984 + 0.702196i 0.752203π0.752203\pi
618618 19.5985 0.788367
619619 27.1282 1.09038 0.545188 0.838314i 0.316458π-0.316458\pi
0.545188 + 0.838314i 0.316458π0.316458\pi
620620 0.955395 0.0383696
621621 −20.2470 −0.812483
622622 −34.6679 −1.39005
623623 0 0
624624 −7.96077 −0.318686
625625 0.772398 0.0308959
626626 11.6993 0.467599
627627 1.71751 0.0685908
628628 −21.2054 −0.846186
629629 −19.9584 −0.795793
630630 0 0
631631 −43.8418 −1.74531 −0.872656 0.488335i 0.837605π-0.837605\pi
−0.872656 + 0.488335i 0.837605π0.837605\pi
632632 8.86294 0.352549
633633 34.6504 1.37723
634634 6.82908 0.271218
635635 14.8562 0.589552
636636 −16.0707 −0.637244
637637 0 0
638638 1.35690 0.0537200
639639 −15.2741 −0.604235
640640 −1.35690 −0.0536360
641641 25.4765 1.00626 0.503131 0.864210i 0.332181π-0.332181\pi
0.503131 + 0.864210i 0.332181π0.332181\pi
642642 −21.7356 −0.857834
643643 −4.20344 −0.165767 −0.0828837 0.996559i 0.526413π-0.526413\pi
−0.0828837 + 0.996559i 0.526413π0.526413\pi
644644 0 0
645645 8.31096 0.327244
646646 12.1685 0.478764
647647 −36.4282 −1.43214 −0.716070 0.698029i 0.754061π-0.754061\pi
−0.716070 + 0.698029i 0.754061π0.754061\pi
648648 2.57673 0.101223
649649 −3.03577 −0.119164
650650 20.1661 0.790981
651651 0 0
652652 18.4862 0.723975
653653 −36.0538 −1.41090 −0.705448 0.708762i 0.749254π-0.749254\pi
−0.705448 + 0.708762i 0.749254π0.749254\pi
654654 8.84415 0.345834
655655 −6.02416 −0.235384
656656 6.60388 0.257838
657657 −14.4819 −0.564992
658658 0 0
659659 −29.5881 −1.15259 −0.576294 0.817243i 0.695502π-0.695502\pi
−0.576294 + 0.817243i 0.695502π0.695502\pi
660660 0.567040 0.0220720
661661 −13.3405 −0.518885 −0.259443 0.965759i 0.583539π-0.583539\pi
−0.259443 + 0.965759i 0.583539π0.583539\pi
662662 15.2664 0.593344
663663 23.5700 0.915384
664664 17.4426 0.676906
665665 0 0
666666 9.74094 0.377454
667667 −14.7899 −0.572666
668668 −20.2741 −0.784430
669669 15.8388 0.612362
670670 −4.27306 −0.165083
671671 −4.40986 −0.170241
672672 0 0
673673 22.5332 0.868591 0.434295 0.900771i 0.356997π-0.356997\pi
0.434295 + 0.900771i 0.356997π0.356997\pi
674674 0.0163935 0.000631452 0
675675 −17.5090 −0.673923
676676 27.7560 1.06754
677677 25.5924 0.983595 0.491798 0.870710i 0.336340π-0.336340\pi
0.491798 + 0.870710i 0.336340π0.336340\pi
678678 −16.1400 −0.619855
679679 0 0
680680 4.01746 0.154062
681681 8.10082 0.310424
682682 0.235963 0.00903549
683683 −2.17390 −0.0831819 −0.0415910 0.999135i 0.513243π-0.513243\pi
−0.0415910 + 0.999135i 0.513243π0.513243\pi
684684 −5.93900 −0.227083
685685 −21.3884 −0.817207
686686 0 0
687687 −12.0978 −0.461561
688688 −4.91185 −0.187263
689689 82.2756 3.13445
690690 −6.18060 −0.235292
691691 −20.5515 −0.781816 −0.390908 0.920430i 0.627839π-0.627839\pi
−0.390908 + 0.920430i 0.627839π0.627839\pi
692692 −6.35690 −0.241653
693693 0 0
694694 −19.5133 −0.740716
695695 −24.3676 −0.924318
696696 5.04892 0.191379
697697 −19.5526 −0.740606
698698 −11.7778 −0.445795
699699 −2.26875 −0.0858120
700700 0 0
701701 20.8170 0.786247 0.393124 0.919486i 0.371394π-0.371394\pi
0.393124 + 0.919486i 0.371394π0.371394\pi
702702 −35.3860 −1.33556
703703 27.7047 1.04490
704704 −0.335126 −0.0126305
705705 1.46011 0.0549908
706706 −34.1672 −1.28590
707707 0 0
708708 −11.2959 −0.424526
709709 8.66679 0.325488 0.162744 0.986668i 0.447965π-0.447965\pi
0.162744 + 0.986668i 0.447965π0.447965\pi
710710 −14.3424 −0.538261
711711 12.8073 0.480312
712712 0.625646 0.0234471
713713 −2.57194 −0.0963200
714714 0 0
715715 −2.90302 −0.108567
716716 3.46011 0.129310
717717 −10.2198 −0.381667
718718 −10.6558 −0.397670
719719 −26.3545 −0.982857 −0.491429 0.870918i 0.663525π-0.663525\pi
−0.491429 + 0.870918i 0.663525π0.663525\pi
720720 −1.96077 −0.0730736
721721 0 0
722722 2.10859 0.0784735
723723 −6.53079 −0.242883
724724 13.7071 0.509420
725725 −12.7899 −0.475003
726726 −13.5767 −0.503880
727727 16.8931 0.626529 0.313265 0.949666i 0.398577π-0.398577\pi
0.313265 + 0.949666i 0.398577π0.398577\pi
728728 0 0
729729 24.4590 0.905890
730730 −13.5985 −0.503303
731731 14.5429 0.537888
732732 −16.4088 −0.606487
733733 17.0586 0.630074 0.315037 0.949079i 0.397983π-0.397983\pi
0.315037 + 0.949079i 0.397983π0.397983\pi
734734 33.0978 1.22166
735735 0 0
736736 3.65279 0.134644
737737 −1.05536 −0.0388747
738738 9.54288 0.351278
739739 −36.3443 −1.33695 −0.668474 0.743735i 0.733052π-0.733052\pi
−0.668474 + 0.743735i 0.733052π0.733052\pi
740740 9.14675 0.336241
741741 −32.7181 −1.20193
742742 0 0
743743 34.1890 1.25427 0.627136 0.778910i 0.284227π-0.284227\pi
0.627136 + 0.778910i 0.284227π0.284227\pi
744744 0.878002 0.0321891
745745 19.1202 0.700510
746746 18.5754 0.680094
747747 25.2054 0.922216
748748 0.992230 0.0362795
749749 0 0
750750 −13.8049 −0.504084
751751 −9.66547 −0.352698 −0.176349 0.984328i 0.556429π-0.556429\pi
−0.176349 + 0.984328i 0.556429π0.556429\pi
752752 −0.862937 −0.0314681
753753 −1.21313 −0.0442088
754754 −25.8485 −0.941345
755755 16.6310 0.605265
756756 0 0
757757 2.97584 0.108159 0.0540793 0.998537i 0.482778π-0.482778\pi
0.0540793 + 0.998537i 0.482778π0.482778\pi
758758 −8.85517 −0.321634
759759 −1.52648 −0.0554078
760760 −5.57673 −0.202289
761761 22.1884 0.804328 0.402164 0.915568i 0.368258π-0.368258\pi
0.402164 + 0.915568i 0.368258π0.368258\pi
762762 13.6528 0.494588
763763 0 0
764764 −6.92931 −0.250694
765765 5.80540 0.209895
766766 30.6969 1.10912
767767 57.8305 2.08814
768768 −1.24698 −0.0449965
769769 −27.5803 −0.994571 −0.497286 0.867587i 0.665670π-0.665670\pi
−0.497286 + 0.867587i 0.665670π0.665670\pi
770770 0 0
771771 19.5985 0.705823
772772 −0.716185 −0.0257760
773773 −36.5327 −1.31399 −0.656995 0.753895i 0.728173π-0.728173\pi
−0.656995 + 0.753895i 0.728173π0.728173\pi
774774 −7.09783 −0.255126
775775 −2.22414 −0.0798936
776776 −3.24698 −0.116560
777777 0 0
778778 1.26205 0.0452465
779779 27.1414 0.972441
780780 −10.8019 −0.386771
781781 −3.54229 −0.126753
782782 −10.8151 −0.386747
783783 22.4426 0.802035
784784 0 0
785785 −28.7735 −1.02697
786786 −5.53617 −0.197469
787787 −14.3870 −0.512842 −0.256421 0.966565i 0.582543π-0.582543\pi
−0.256421 + 0.966565i 0.582543π0.582543\pi
788788 2.06100 0.0734200
789789 23.1468 0.824046
790790 12.0261 0.427869
791791 0 0
792792 −0.484271 −0.0172078
793793 84.0066 2.98316
794794 12.4558 0.442040
795795 −21.8062 −0.773388
796796 0.0217703 0.000771627 0
797797 −19.1564 −0.678556 −0.339278 0.940686i 0.610183π-0.610183\pi
−0.339278 + 0.940686i 0.610183π0.610183\pi
798798 0 0
799799 2.55496 0.0903879
800800 3.15883 0.111682
801801 0.904084 0.0319442
802802 11.8267 0.417615
803803 −3.35855 −0.118521
804804 −3.92692 −0.138492
805805 0 0
806806 −4.49502 −0.158330
807807 −13.8793 −0.488576
808808 −1.77479 −0.0624369
809809 −47.2771 −1.66217 −0.831087 0.556142i 0.812281π-0.812281\pi
−0.831087 + 0.556142i 0.812281π0.812281\pi
810810 3.49635 0.122849
811811 35.9691 1.26305 0.631524 0.775357i 0.282430π-0.282430\pi
0.631524 + 0.775357i 0.282430π0.282430\pi
812812 0 0
813813 −25.0954 −0.880136
814814 2.25906 0.0791801
815815 25.0838 0.878648
816816 3.69202 0.129247
817817 −20.1873 −0.706265
818818 −18.4058 −0.643544
819819 0 0
820820 8.96077 0.312924
821821 −15.9554 −0.556847 −0.278424 0.960458i 0.589812π-0.589812\pi
−0.278424 + 0.960458i 0.589812π0.589812\pi
822822 −19.6558 −0.685574
823823 32.5810 1.13570 0.567852 0.823131i 0.307775π-0.307775\pi
0.567852 + 0.823131i 0.307775π0.307775\pi
824824 −15.7168 −0.547520
825825 −1.32006 −0.0459586
826826 0 0
827827 18.6300 0.647827 0.323914 0.946087i 0.395001π-0.395001\pi
0.323914 + 0.946087i 0.395001π0.395001\pi
828828 5.27844 0.183438
829829 −16.9661 −0.589259 −0.294629 0.955612i 0.595196π-0.595196\pi
−0.294629 + 0.955612i 0.595196π0.595196\pi
830830 23.6679 0.821523
831831 32.2325 1.11813
832832 6.38404 0.221327
833833 0 0
834834 −22.3937 −0.775432
835835 −27.5099 −0.952019
836836 −1.37734 −0.0476362
837837 3.90276 0.134899
838838 5.12067 0.176891
839839 −9.12365 −0.314984 −0.157492 0.987520i 0.550341π-0.550341\pi
−0.157492 + 0.987520i 0.550341π0.550341\pi
840840 0 0
841841 −12.6063 −0.434699
842842 16.4034 0.565299
843843 16.3026 0.561491
844844 −27.7875 −0.956484
845845 37.6620 1.29561
846846 −1.24698 −0.0428720
847847 0 0
848848 12.8877 0.442565
849849 −7.20403 −0.247242
850850 −9.35258 −0.320791
851851 −24.6233 −0.844074
852852 −13.1806 −0.451560
853853 −10.2620 −0.351366 −0.175683 0.984447i 0.556213π-0.556213\pi
−0.175683 + 0.984447i 0.556213π0.556213\pi
854854 0 0
855855 −8.05861 −0.275599
856856 17.4306 0.595765
857857 11.2832 0.385428 0.192714 0.981255i 0.438271π-0.438271\pi
0.192714 + 0.981255i 0.438271π0.438271\pi
858858 −2.66786 −0.0910792
859859 20.0242 0.683216 0.341608 0.939843i 0.389029π-0.389029\pi
0.341608 + 0.939843i 0.389029π0.389029\pi
860860 −6.66487 −0.227270
861861 0 0
862862 0.330814 0.0112676
863863 41.4596 1.41130 0.705651 0.708559i 0.250655π-0.250655\pi
0.705651 + 0.708559i 0.250655π0.250655\pi
864864 −5.54288 −0.188572
865865 −8.62565 −0.293281
866866 2.45042 0.0832686
867867 10.2674 0.348700
868868 0 0
869869 2.97020 0.100757
870870 6.85086 0.232266
871871 20.1043 0.681207
872872 −7.09246 −0.240181
873873 −4.69202 −0.158801
874874 15.0127 0.507811
875875 0 0
876876 −12.4969 −0.422233
877877 24.2034 0.817292 0.408646 0.912693i 0.366001π-0.366001\pi
0.408646 + 0.912693i 0.366001π0.366001\pi
878878 −2.69202 −0.0908513
879879 −25.5881 −0.863064
880880 −0.454731 −0.0153290
881881 37.3793 1.25934 0.629670 0.776863i 0.283190π-0.283190\pi
0.629670 + 0.776863i 0.283190π0.283190\pi
882882 0 0
883883 0.550172 0.0185148 0.00925739 0.999957i 0.497053π-0.497053\pi
0.00925739 + 0.999957i 0.497053π0.497053\pi
884884 −18.9017 −0.635733
885885 −15.3274 −0.515223
886886 −36.5526 −1.22801
887887 25.3521 0.851241 0.425620 0.904902i 0.360056π-0.360056\pi
0.425620 + 0.904902i 0.360056π0.360056\pi
888888 8.40581 0.282081
889889 0 0
890890 0.848936 0.0284564
891891 0.863528 0.0289293
892892 −12.7017 −0.425285
893893 −3.54660 −0.118682
894894 17.5714 0.587674
895895 4.69501 0.156937
896896 0 0
897897 29.0790 0.970921
898898 3.48427 0.116272
899899 2.85086 0.0950813
900900 4.56465 0.152155
901901 −38.1575 −1.27121
902902 2.21313 0.0736891
903903 0 0
904904 12.9433 0.430488
905905 18.5991 0.618255
906906 15.2838 0.507771
907907 53.9512 1.79142 0.895710 0.444639i 0.146668π-0.146668\pi
0.895710 + 0.444639i 0.146668π0.146668\pi
908908 −6.49635 −0.215589
909909 −2.56465 −0.0850640
910910 0 0
911911 13.2433 0.438769 0.219384 0.975639i 0.429595π-0.429595\pi
0.219384 + 0.975639i 0.429595π0.429595\pi
912912 −5.12498 −0.169705
913913 5.84548 0.193457
914914 23.1183 0.764685
915915 −22.2650 −0.736059
916916 9.70171 0.320554
917917 0 0
918918 16.4112 0.541650
919919 −24.2389 −0.799569 −0.399785 0.916609i 0.630915π-0.630915\pi
−0.399785 + 0.916609i 0.630915π0.630915\pi
920920 4.95646 0.163410
921921 −12.7748 −0.420944
922922 −9.69501 −0.319288
923923 67.4795 2.22111
924924 0 0
925925 −21.2935 −0.700126
926926 5.84654 0.192129
927927 −22.7114 −0.745940
928928 −4.04892 −0.132912
929929 −12.4295 −0.407799 −0.203899 0.978992i 0.565362π-0.565362\pi
−0.203899 + 0.978992i 0.565362π0.565362\pi
930930 1.19136 0.0390662
931931 0 0
932932 1.81940 0.0595963
933933 −43.2301 −1.41529
934934 25.7832 0.843650
935935 1.34635 0.0440305
936936 9.22521 0.301535
937937 25.0672 0.818911 0.409455 0.912330i 0.365719π-0.365719\pi
0.409455 + 0.912330i 0.365719π0.365719\pi
938938 0 0
939939 14.5888 0.476088
940940 −1.17092 −0.0381910
941941 −41.8961 −1.36577 −0.682886 0.730525i 0.739275π-0.739275\pi
−0.682886 + 0.730525i 0.739275π0.739275\pi
942942 −26.4426 −0.861548
943943 −24.1226 −0.785540
944944 9.05861 0.294833
945945 0 0
946946 −1.64609 −0.0535189
947947 13.9788 0.454251 0.227125 0.973866i 0.427067π-0.427067\pi
0.227125 + 0.973866i 0.427067π0.427067\pi
948948 11.0519 0.358949
949949 63.9794 2.07686
950950 12.9825 0.421209
951951 8.51573 0.276141
952952 0 0
953953 −14.9038 −0.482782 −0.241391 0.970428i 0.577604π-0.577604\pi
−0.241391 + 0.970428i 0.577604π0.577604\pi
954954 18.6233 0.602950
955955 −9.40236 −0.304253
956956 8.19567 0.265067
957957 1.69202 0.0546953
958958 −20.8799 −0.674600
959959 0 0
960960 −1.69202 −0.0546098
961961 −30.5042 −0.984008
962962 −43.0344 −1.38749
963963 25.1879 0.811669
964964 5.23729 0.168682
965965 −0.971788 −0.0312830
966966 0 0
967967 42.6469 1.37143 0.685717 0.727869i 0.259489π-0.259489\pi
0.685717 + 0.727869i 0.259489π0.259489\pi
968968 10.8877 0.349944
969969 15.1739 0.487456
970970 −4.40581 −0.141462
971971 −3.03790 −0.0974909 −0.0487454 0.998811i 0.515522π-0.515522\pi
−0.0487454 + 0.998811i 0.515522π0.515522\pi
972972 −13.4155 −0.430302
973973 0 0
974974 0.629958 0.0201851
975975 25.1468 0.805341
976976 13.1588 0.421204
977977 5.60089 0.179188 0.0895942 0.995978i 0.471443π-0.471443\pi
0.0895942 + 0.995978i 0.471443π0.471443\pi
978978 23.0519 0.737119
979979 0.209670 0.00670108
980980 0 0
981981 −10.2489 −0.327222
982982 7.19269 0.229528
983983 −8.13600 −0.259498 −0.129749 0.991547i 0.541417π-0.541417\pi
−0.129749 + 0.991547i 0.541417π0.541417\pi
984984 8.23490 0.262519
985985 2.79656 0.0891058
986986 11.9879 0.381773
987987 0 0
988988 26.2379 0.834738
989989 17.9420 0.570522
990990 −0.657105 −0.0208842
991991 −16.1637 −0.513458 −0.256729 0.966483i 0.582645π-0.582645\pi
−0.256729 + 0.966483i 0.582645π0.582645\pi
992992 −0.704103 −0.0223553
993993 19.0368 0.604116
994994 0 0
995995 0.0295400 0.000936480 0
996996 21.7506 0.689195
997997 −31.6614 −1.00273 −0.501364 0.865237i 0.667168π-0.667168\pi
−0.501364 + 0.865237i 0.667168π0.667168\pi
998998 −21.2567 −0.672868
999999 37.3642 1.18215
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 686.2.a.b.1.1 yes 3
3.2 odd 2 6174.2.a.l.1.2 3
4.3 odd 2 5488.2.a.b.1.3 3
7.2 even 3 686.2.c.c.361.3 6
7.3 odd 6 686.2.c.d.667.1 6
7.4 even 3 686.2.c.c.667.3 6
7.5 odd 6 686.2.c.d.361.1 6
7.6 odd 2 686.2.a.a.1.3 3
21.20 even 2 6174.2.a.k.1.2 3
28.27 even 2 5488.2.a.e.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
686.2.a.a.1.3 3 7.6 odd 2
686.2.a.b.1.1 yes 3 1.1 even 1 trivial
686.2.c.c.361.3 6 7.2 even 3
686.2.c.c.667.3 6 7.4 even 3
686.2.c.d.361.1 6 7.5 odd 6
686.2.c.d.667.1 6 7.3 odd 6
5488.2.a.b.1.3 3 4.3 odd 2
5488.2.a.e.1.1 3 28.27 even 2
6174.2.a.k.1.2 3 21.20 even 2
6174.2.a.l.1.2 3 3.2 odd 2