Properties

Label 45.3.c.a.26.1
Level 4545
Weight 33
Character 45.26
Analytic conductor 1.2261.226
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,3,Mod(26,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: N N == 45=325 45 = 3^{2} \cdot 5
Weight: k k == 3 3
Character orbit: [χ][\chi] == 45.c (of order 22, degree 11, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 1.226161189621.22616118962
Analytic rank: 00
Dimension: 44
Coefficient field: Q(2,5)\Q(\sqrt{-2}, \sqrt{-5})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x44x2+9 x^{4} - 4x^{2} + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 2 2
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 26.1
Root 1.58114+0.707107i1.58114 + 0.707107i of defining polynomial
Character χ\chi == 45.26
Dual form 45.3.c.a.26.4

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) == q3.65028iq29.32456q42.23607iq5+7.16228q7+19.4361iq88.16228q105.42736iq11+9.81139q1326.1443iq14+33.6491q16+12.2317iq17+6.32456q19+20.8503iq2019.8114q22+12.0394iq235.00000q2535.8143iq2666.7851q28+44.9881iq2958.2719q3145.0842iq32+44.6491q3416.0153iq35+66.4605q3723.0864iq38+43.4605q4016.4743iq4143.6228q43+50.6077iq44+43.9473q4640.0570iq47+2.29822q49+18.2514iq5091.4868q5213.2242iq5312.1359q55+139.207iq56+164.219q5825.1519iq5935.6754q61+212.709iq6229.9737q6421.9389iq65+26.7018q67114.055iq6858.4605q7092.7301iq71+60.3246q73242.600iq7458.9737q7638.8723iq7796.2192q7975.2417iq8060.1359q82+79.1215iq83+27.3509q85+159.235iq86+105.487q88+107.443iq89+70.2719q91112.262iq92146.219q9414.1421iq951.07900q978.38915iq98+O(q100)q-3.65028i q^{2} -9.32456 q^{4} -2.23607i q^{5} +7.16228 q^{7} +19.4361i q^{8} -8.16228 q^{10} -5.42736i q^{11} +9.81139 q^{13} -26.1443i q^{14} +33.6491 q^{16} +12.2317i q^{17} +6.32456 q^{19} +20.8503i q^{20} -19.8114 q^{22} +12.0394i q^{23} -5.00000 q^{25} -35.8143i q^{26} -66.7851 q^{28} +44.9881i q^{29} -58.2719 q^{31} -45.0842i q^{32} +44.6491 q^{34} -16.0153i q^{35} +66.4605 q^{37} -23.0864i q^{38} +43.4605 q^{40} -16.4743i q^{41} -43.6228 q^{43} +50.6077i q^{44} +43.9473 q^{46} -40.0570i q^{47} +2.29822 q^{49} +18.2514i q^{50} -91.4868 q^{52} -13.2242i q^{53} -12.1359 q^{55} +139.207i q^{56} +164.219 q^{58} -25.1519i q^{59} -35.6754 q^{61} +212.709i q^{62} -29.9737 q^{64} -21.9389i q^{65} +26.7018 q^{67} -114.055i q^{68} -58.4605 q^{70} -92.7301i q^{71} +60.3246 q^{73} -242.600i q^{74} -58.9737 q^{76} -38.8723i q^{77} -96.2192 q^{79} -75.2417i q^{80} -60.1359 q^{82} +79.1215i q^{83} +27.3509 q^{85} +159.235i q^{86} +105.487 q^{88} +107.443i q^{89} +70.2719 q^{91} -112.262i q^{92} -146.219 q^{94} -14.1421i q^{95} -1.07900 q^{97} -8.38915i q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q12q4+16q720q1024q13+84q1616q2220q25128q2856q31+128q34+152q37+60q4048q43+24q4692q49328q52+40q55+232q97+O(q100) 4 q - 12 q^{4} + 16 q^{7} - 20 q^{10} - 24 q^{13} + 84 q^{16} - 16 q^{22} - 20 q^{25} - 128 q^{28} - 56 q^{31} + 128 q^{34} + 152 q^{37} + 60 q^{40} - 48 q^{43} + 24 q^{46} - 92 q^{49} - 328 q^{52} + 40 q^{55}+ \cdots - 232 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/45Z)×\left(\mathbb{Z}/45\mathbb{Z}\right)^\times.

nn 1111 3737
χ(n)\chi(n) 1-1 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 − 3.65028i − 1.82514i −0.408920 0.912570i 0.634094π-0.634094\pi
0.408920 0.912570i 0.365906π-0.365906\pi
33 0 0
44 −9.32456 −2.33114
55 − 2.23607i − 0.447214i
66 0 0
77 7.16228 1.02318 0.511591 0.859229i 0.329056π-0.329056\pi
0.511591 + 0.859229i 0.329056π0.329056\pi
88 19.4361i 2.42952i
99 0 0
1010 −8.16228 −0.816228
1111 − 5.42736i − 0.493396i −0.969092 0.246698i 0.920654π-0.920654\pi
0.969092 0.246698i 0.0793456π-0.0793456\pi
1212 0 0
1313 9.81139 0.754722 0.377361 0.926066i 0.376832π-0.376832\pi
0.377361 + 0.926066i 0.376832π0.376832\pi
1414 − 26.1443i − 1.86745i
1515 0 0
1616 33.6491 2.10307
1717 12.2317i 0.719511i 0.933047 + 0.359756i 0.117140π0.117140\pi
−0.933047 + 0.359756i 0.882860π0.882860\pi
1818 0 0
1919 6.32456 0.332871 0.166436 0.986052i 0.446774π-0.446774\pi
0.166436 + 0.986052i 0.446774π0.446774\pi
2020 20.8503i 1.04252i
2121 0 0
2222 −19.8114 −0.900518
2323 12.0394i 0.523454i 0.965142 + 0.261727i 0.0842920π0.0842920\pi
−0.965142 + 0.261727i 0.915708π0.915708\pi
2424 0 0
2525 −5.00000 −0.200000
2626 − 35.8143i − 1.37747i
2727 0 0
2828 −66.7851 −2.38518
2929 44.9881i 1.55131i 0.631155 + 0.775657i 0.282581π0.282581\pi
−0.631155 + 0.775657i 0.717419π0.717419\pi
3030 0 0
3131 −58.2719 −1.87974 −0.939869 0.341535i 0.889053π-0.889053\pi
−0.939869 + 0.341535i 0.889053π0.889053\pi
3232 − 45.0842i − 1.40888i
3333 0 0
3434 44.6491 1.31321
3535 − 16.0153i − 0.457581i
3636 0 0
3737 66.4605 1.79623 0.898115 0.439761i 0.144937π-0.144937\pi
0.898115 + 0.439761i 0.144937π0.144937\pi
3838 − 23.0864i − 0.607537i
3939 0 0
4040 43.4605 1.08651
4141 − 16.4743i − 0.401813i −0.979610 0.200906i 0.935611π-0.935611\pi
0.979610 0.200906i 0.0643887π-0.0643887\pi
4242 0 0
4343 −43.6228 −1.01448 −0.507242 0.861804i 0.669335π-0.669335\pi
−0.507242 + 0.861804i 0.669335π0.669335\pi
4444 50.6077i 1.15018i
4545 0 0
4646 43.9473 0.955377
4747 − 40.0570i − 0.852276i −0.904658 0.426138i 0.859874π-0.859874\pi
0.904658 0.426138i 0.140126π-0.140126\pi
4848 0 0
4949 2.29822 0.0469025
5050 18.2514i 0.365028i
5151 0 0
5252 −91.4868 −1.75936
5353 − 13.2242i − 0.249512i −0.992187 0.124756i 0.960185π-0.960185\pi
0.992187 0.124756i 0.0398149π-0.0398149\pi
5454 0 0
5555 −12.1359 −0.220654
5656 139.207i 2.48584i
5757 0 0
5858 164.219 2.83137
5959 − 25.1519i − 0.426303i −0.977019 0.213151i 0.931627π-0.931627\pi
0.977019 0.213151i 0.0683728π-0.0683728\pi
6060 0 0
6161 −35.6754 −0.584843 −0.292422 0.956289i 0.594461π-0.594461\pi
−0.292422 + 0.956289i 0.594461π0.594461\pi
6262 212.709i 3.43079i
6363 0 0
6464 −29.9737 −0.468339
6565 − 21.9389i − 0.337522i
6666 0 0
6767 26.7018 0.398534 0.199267 0.979945i 0.436144π-0.436144\pi
0.199267 + 0.979945i 0.436144π0.436144\pi
6868 − 114.055i − 1.67728i
6969 0 0
7070 −58.4605 −0.835150
7171 − 92.7301i − 1.30606i −0.757333 0.653029i 0.773498π-0.773498\pi
0.757333 0.653029i 0.226502π-0.226502\pi
7272 0 0
7373 60.3246 0.826364 0.413182 0.910649i 0.364417π-0.364417\pi
0.413182 + 0.910649i 0.364417π0.364417\pi
7474 − 242.600i − 3.27837i
7575 0 0
7676 −58.9737 −0.775969
7777 − 38.8723i − 0.504834i
7878 0 0
7979 −96.2192 −1.21796 −0.608982 0.793184i 0.708422π-0.708422\pi
−0.608982 + 0.793184i 0.708422π0.708422\pi
8080 − 75.2417i − 0.940521i
8181 0 0
8282 −60.1359 −0.733365
8383 79.1215i 0.953271i 0.879101 + 0.476635i 0.158144π0.158144\pi
−0.879101 + 0.476635i 0.841856π0.841856\pi
8484 0 0
8585 27.3509 0.321775
8686 159.235i 1.85157i
8787 0 0
8888 105.487 1.19871
8989 107.443i 1.20722i 0.797278 + 0.603612i 0.206272π0.206272\pi
−0.797278 + 0.603612i 0.793728π0.793728\pi
9090 0 0
9191 70.2719 0.772219
9292 − 112.262i − 1.22024i
9393 0 0
9494 −146.219 −1.55552
9595 − 14.1421i − 0.148865i
9696 0 0
9797 −1.07900 −0.0111237 −0.00556187 0.999985i 0.501770π-0.501770\pi
−0.00556187 + 0.999985i 0.501770π0.501770\pi
9898 − 8.38915i − 0.0856036i
9999 0 0
100100 46.6228 0.466228
101101 170.282i 1.68596i 0.537942 + 0.842982i 0.319202π0.319202\pi
−0.537942 + 0.842982i 0.680798π0.680798\pi
102102 0 0
103103 −128.460 −1.24719 −0.623595 0.781748i 0.714328π-0.714328\pi
−0.623595 + 0.781748i 0.714328π0.714328\pi
104104 190.695i 1.83361i
105105 0 0
106106 −48.2719 −0.455395
107107 − 76.3675i − 0.713715i −0.934159 0.356858i 0.883848π-0.883848\pi
0.934159 0.356858i 0.116152π-0.116152\pi
108108 0 0
109109 −13.0790 −0.119991 −0.0599954 0.998199i 0.519109π-0.519109\pi
−0.0599954 + 0.998199i 0.519109π0.519109\pi
110110 44.2996i 0.402724i
111111 0 0
112112 241.004 2.15182
113113 20.7170i 0.183336i 0.995790 + 0.0916680i 0.0292199π0.0292199\pi
−0.995790 + 0.0916680i 0.970780π0.970780\pi
114114 0 0
115115 26.9210 0.234096
116116 − 419.494i − 3.61633i
117117 0 0
118118 −91.8114 −0.778063
119119 87.6068i 0.736191i
120120 0 0
121121 91.5438 0.756560
122122 130.225i 1.06742i
123123 0 0
124124 543.359 4.38193
125125 11.1803i 0.0894427i
126126 0 0
127127 −38.0306 −0.299454 −0.149727 0.988727i 0.547839π-0.547839\pi
−0.149727 + 0.988727i 0.547839π0.547839\pi
128128 − 70.9246i − 0.554098i
129129 0 0
130130 −80.0833 −0.616025
131131 83.9409i 0.640770i 0.947287 + 0.320385i 0.103812π0.103812\pi
−0.947287 + 0.320385i 0.896188π0.896188\pi
132132 0 0
133133 45.2982 0.340588
134134 − 97.4690i − 0.727381i
135135 0 0
136136 −237.737 −1.74806
137137 − 15.5936i − 0.113822i −0.998379 0.0569109i 0.981875π-0.981875\pi
0.998379 0.0569109i 0.0181251π-0.0181251\pi
138138 0 0
139139 67.8420 0.488072 0.244036 0.969766i 0.421529π-0.421529\pi
0.244036 + 0.969766i 0.421529π0.421529\pi
140140 149.336i 1.06669i
141141 0 0
142142 −338.491 −2.38374
143143 − 53.2499i − 0.372377i
144144 0 0
145145 100.596 0.693769
146146 − 220.202i − 1.50823i
147147 0 0
148148 −619.715 −4.18726
149149 − 233.426i − 1.56662i −0.621634 0.783308i 0.713531π-0.713531\pi
0.621634 0.783308i 0.286469π-0.286469\pi
150150 0 0
151151 185.351 1.22749 0.613745 0.789505i 0.289662π-0.289662\pi
0.613745 + 0.789505i 0.289662π0.289662\pi
152152 122.925i 0.808716i
153153 0 0
154154 −141.895 −0.921394
155155 130.300i 0.840645i
156156 0 0
157157 −111.276 −0.708765 −0.354383 0.935100i 0.615309π-0.615309\pi
−0.354383 + 0.935100i 0.615309π0.615309\pi
158158 351.227i 2.22296i
159159 0 0
160160 −100.811 −0.630071
161161 86.2298i 0.535589i
162162 0 0
163163 118.763 0.728607 0.364304 0.931280i 0.381307π-0.381307\pi
0.364304 + 0.931280i 0.381307π0.381307\pi
164164 153.616i 0.936682i
165165 0 0
166166 288.816 1.73985
167167 − 221.194i − 1.32452i −0.749276 0.662258i 0.769598π-0.769598\pi
0.749276 0.662258i 0.230402π-0.230402\pi
168168 0 0
169169 −72.7367 −0.430394
170170 − 99.8384i − 0.587285i
171171 0 0
172172 406.763 2.36490
173173 − 190.807i − 1.10293i −0.834198 0.551466i 0.814069π-0.814069\pi
0.834198 0.551466i 0.185931π-0.185931\pi
174174 0 0
175175 −35.8114 −0.204637
176176 − 182.626i − 1.03765i
177177 0 0
178178 392.197 2.20335
179179 58.1005i 0.324584i 0.986743 + 0.162292i 0.0518886π0.0518886\pi
−0.986743 + 0.162292i 0.948111π0.948111\pi
180180 0 0
181181 −162.921 −0.900116 −0.450058 0.892999i 0.648597π-0.648597\pi
−0.450058 + 0.892999i 0.648597π0.648597\pi
182182 − 256.512i − 1.40941i
183183 0 0
184184 −234.000 −1.27174
185185 − 148.610i − 0.803298i
186186 0 0
187187 66.3858 0.355004
188188 373.513i 1.98677i
189189 0 0
190190 −51.6228 −0.271699
191191 − 100.062i − 0.523884i −0.965084 0.261942i 0.915637π-0.915637\pi
0.965084 0.261942i 0.0843630π-0.0843630\pi
192192 0 0
193193 −61.8947 −0.320698 −0.160349 0.987060i 0.551262π-0.551262\pi
−0.160349 + 0.987060i 0.551262π0.551262\pi
194194 3.93866i 0.0203024i
195195 0 0
196196 −21.4299 −0.109336
197197 24.8791i 0.126290i 0.998004 + 0.0631449i 0.0201130π0.0201130\pi
−0.998004 + 0.0631449i 0.979887π0.979887\pi
198198 0 0
199199 −156.491 −0.786387 −0.393194 0.919456i 0.628630π-0.628630\pi
−0.393194 + 0.919456i 0.628630π0.628630\pi
200200 − 97.1806i − 0.485903i
201201 0 0
202202 621.579 3.07712
203203 322.217i 1.58728i
204204 0 0
205205 −36.8377 −0.179696
206206 468.917i 2.27630i
207207 0 0
208208 330.144 1.58723
209209 − 34.3256i − 0.164237i
210210 0 0
211211 237.789 1.12696 0.563482 0.826128i 0.309461π-0.309461\pi
0.563482 + 0.826128i 0.309461π0.309461\pi
212212 123.309i 0.581648i
213213 0 0
214214 −278.763 −1.30263
215215 97.5435i 0.453691i
216216 0 0
217217 −417.359 −1.92332
218218 47.7420i 0.219000i
219219 0 0
220220 113.162 0.514374
221221 120.010i 0.543031i
222222 0 0
223223 −182.302 −0.817500 −0.408750 0.912646i 0.634035π-0.634035\pi
−0.408750 + 0.912646i 0.634035π0.634035\pi
224224 − 322.906i − 1.44154i
225225 0 0
226226 75.6228 0.334614
227227 406.078i 1.78889i 0.447180 + 0.894444i 0.352428π0.352428\pi
−0.447180 + 0.894444i 0.647572π0.647572\pi
228228 0 0
229229 −27.2982 −0.119206 −0.0596031 0.998222i 0.518984π-0.518984\pi
−0.0596031 + 0.998222i 0.518984π0.518984\pi
230230 − 98.2692i − 0.427257i
231231 0 0
232232 −874.394 −3.76894
233233 − 356.382i − 1.52954i −0.644306 0.764768i 0.722854π-0.722854\pi
0.644306 0.764768i 0.277146π-0.277146\pi
234234 0 0
235235 −89.5701 −0.381149
236236 234.530i 0.993771i
237237 0 0
238238 319.789 1.34365
239239 − 271.690i − 1.13678i −0.822760 0.568389i 0.807567π-0.807567\pi
0.822760 0.568389i 0.192433π-0.192433\pi
240240 0 0
241241 −224.438 −0.931280 −0.465640 0.884974i 0.654176π-0.654176\pi
−0.465640 + 0.884974i 0.654176π0.654176\pi
242242 − 334.161i − 1.38083i
243243 0 0
244244 332.658 1.36335
245245 − 5.13898i − 0.0209754i
246246 0 0
247247 62.0527 0.251225
248248 − 1132.58i − 4.56685i
249249 0 0
250250 40.8114 0.163246
251251 − 318.775i − 1.27002i −0.772504 0.635010i 0.780996π-0.780996\pi
0.772504 0.635010i 0.219004π-0.219004\pi
252252 0 0
253253 65.3423 0.258270
254254 138.822i 0.546545i
255255 0 0
256256 −378.789 −1.47965
257257 371.975i 1.44738i 0.690128 + 0.723688i 0.257554π0.257554\pi
−0.690128 + 0.723688i 0.742446π0.742446\pi
258258 0 0
259259 476.009 1.83787
260260 204.571i 0.786811i
261261 0 0
262262 306.408 1.16950
263263 238.549i 0.907031i 0.891248 + 0.453515i 0.149830π0.149830\pi
−0.891248 + 0.453515i 0.850170π0.850170\pi
264264 0 0
265265 −29.5701 −0.111585
266266 − 165.351i − 0.621621i
267267 0 0
268268 −248.982 −0.929038
269269 125.871i 0.467922i 0.972246 + 0.233961i 0.0751688π0.0751688\pi
−0.972246 + 0.233961i 0.924831π0.924831\pi
270270 0 0
271271 −258.649 −0.954425 −0.477212 0.878788i 0.658353π-0.658353\pi
−0.477212 + 0.878788i 0.658353π0.658353\pi
272272 411.585i 1.51318i
273273 0 0
274274 −56.9210 −0.207741
275275 27.1368i 0.0986793i
276276 0 0
277277 227.715 0.822074 0.411037 0.911619i 0.365167π-0.365167\pi
0.411037 + 0.911619i 0.365167π0.365167\pi
278278 − 247.642i − 0.890800i
279279 0 0
280280 311.276 1.11170
281281 − 241.384i − 0.859016i −0.903063 0.429508i 0.858687π-0.858687\pi
0.903063 0.429508i 0.141313π-0.141313\pi
282282 0 0
283283 208.333 0.736159 0.368080 0.929794i 0.380015π-0.380015\pi
0.368080 + 0.929794i 0.380015π0.380015\pi
284284 864.667i 3.04460i
285285 0 0
286286 −194.377 −0.679641
287287 − 117.994i − 0.411128i
288288 0 0
289289 139.386 0.482304
290290 − 367.205i − 1.26623i
291291 0 0
292292 −562.500 −1.92637
293293 − 201.693i − 0.688372i −0.938901 0.344186i 0.888155π-0.888155\pi
0.938901 0.344186i 0.111845π-0.111845\pi
294294 0 0
295295 −56.2413 −0.190648
296296 1291.73i 4.36397i
297297 0 0
298298 −852.070 −2.85929
299299 118.124i 0.395062i
300300 0 0
301301 −312.438 −1.03800
302302 − 676.583i − 2.24034i
303303 0 0
304304 212.816 0.700052
305305 79.7727i 0.261550i
306306 0 0
307307 342.824 1.11669 0.558346 0.829608i 0.311436π-0.311436\pi
0.558346 + 0.829608i 0.311436π0.311436\pi
308308 362.466i 1.17684i
309309 0 0
310310 475.631 1.53429
311311 − 217.640i − 0.699807i −0.936786 0.349903i 0.886214π-0.886214\pi
0.936786 0.349903i 0.113786π-0.113786\pi
312312 0 0
313313 281.895 0.900622 0.450311 0.892872i 0.351313π-0.351313\pi
0.450311 + 0.892872i 0.351313π0.351313\pi
314314 406.189i 1.29360i
315315 0 0
316316 897.201 2.83925
317317 15.4013i 0.0485847i 0.999705 + 0.0242923i 0.00773325π0.00773325\pi
−0.999705 + 0.0242923i 0.992267π0.992267\pi
318318 0 0
319319 244.167 0.765412
320320 67.0232i 0.209447i
321321 0 0
322322 314.763 0.977525
323323 77.3600i 0.239505i
324324 0 0
325325 −49.0569 −0.150944
326326 − 433.518i − 1.32981i
327327 0 0
328328 320.197 0.976211
329329 − 286.899i − 0.872034i
330330 0 0
331331 375.517 1.13449 0.567247 0.823548i 0.308009π-0.308009\pi
0.567247 + 0.823548i 0.308009π0.308009\pi
332332 − 737.773i − 2.22221i
333333 0 0
334334 −807.421 −2.41743
335335 − 59.7070i − 0.178230i
336336 0 0
337337 188.114 0.558201 0.279101 0.960262i 0.409964π-0.409964\pi
0.279101 + 0.960262i 0.409964π0.409964\pi
338338 265.509i 0.785530i
339339 0 0
340340 −255.035 −0.750103
341341 316.262i 0.927456i
342342 0 0
343343 −334.491 −0.975193
344344 − 847.858i − 2.46470i
345345 0 0
346346 −696.500 −2.01300
347347 513.793i 1.48067i 0.672237 + 0.740336i 0.265334π0.265334\pi
−0.672237 + 0.740336i 0.734666π0.734666\pi
348348 0 0
349349 112.535 0.322451 0.161225 0.986918i 0.448455π-0.448455\pi
0.161225 + 0.986918i 0.448455π0.448455\pi
350350 130.722i 0.373490i
351351 0 0
352352 −244.688 −0.695137
353353 − 428.172i − 1.21295i −0.795102 0.606475i 0.792583π-0.792583\pi
0.795102 0.606475i 0.207417π-0.207417\pi
354354 0 0
355355 −207.351 −0.584087
356356 − 1001.86i − 2.81421i
357357 0 0
358358 212.083 0.592411
359359 − 56.1961i − 0.156535i −0.996932 0.0782676i 0.975061π-0.975061\pi
0.996932 0.0782676i 0.0249389π-0.0249389\pi
360360 0 0
361361 −321.000 −0.889197
362362 594.708i 1.64284i
363363 0 0
364364 −655.254 −1.80015
365365 − 134.890i − 0.369561i
366366 0 0
367367 −154.364 −0.420610 −0.210305 0.977636i 0.567446π-0.567446\pi
−0.210305 + 0.977636i 0.567446π0.567446\pi
368368 405.116i 1.10086i
369369 0 0
370370 −542.469 −1.46613
371371 − 94.7151i − 0.255297i
372372 0 0
373373 557.285 1.49406 0.747030 0.664790i 0.231479π-0.231479\pi
0.747030 + 0.664790i 0.231479π0.231479\pi
374374 − 242.327i − 0.647932i
375375 0 0
376376 778.552 2.07062
377377 441.396i 1.17081i
378378 0 0
379379 147.404 0.388928 0.194464 0.980910i 0.437703π-0.437703\pi
0.194464 + 0.980910i 0.437703π0.437703\pi
380380 131.869i 0.347024i
381381 0 0
382382 −365.254 −0.956163
383383 736.619i 1.92329i 0.274302 + 0.961644i 0.411553π0.411553\pi
−0.274302 + 0.961644i 0.588447π0.588447\pi
384384 0 0
385385 −86.9210 −0.225769
386386 225.933i 0.585319i
387387 0 0
388388 10.0612 0.0259310
389389 296.408i 0.761975i 0.924580 + 0.380987i 0.124416π0.124416\pi
−0.924580 + 0.380987i 0.875584π0.875584\pi
390390 0 0
391391 −147.263 −0.376631
392392 44.6685i 0.113950i
393393 0 0
394394 90.8157 0.230497
395395 215.153i 0.544690i
396396 0 0
397397 −457.057 −1.15128 −0.575638 0.817704i 0.695246π-0.695246\pi
−0.575638 + 0.817704i 0.695246π0.695246\pi
398398 571.237i 1.43527i
399399 0 0
400400 −168.246 −0.420614
401401 391.141i 0.975415i 0.873007 + 0.487707i 0.162167π0.162167\pi
−0.873007 + 0.487707i 0.837833π0.837833\pi
402402 0 0
403403 −571.728 −1.41868
404404 − 1587.81i − 3.93022i
405405 0 0
406406 1176.18 2.89700
407407 − 360.705i − 0.886253i
408408 0 0
409409 −411.842 −1.00695 −0.503474 0.864010i 0.667945π-0.667945\pi
−0.503474 + 0.864010i 0.667945π0.667945\pi
410410 134.468i 0.327971i
411411 0 0
412412 1197.84 2.90737
413413 − 180.145i − 0.436186i
414414 0 0
415415 176.921 0.426316
416416 − 442.339i − 1.06331i
417417 0 0
418418 −125.298 −0.299757
419419 − 653.447i − 1.55954i −0.626066 0.779770i 0.715336π-0.715336\pi
0.626066 0.779770i 0.284664π-0.284664\pi
420420 0 0
421421 125.035 0.296995 0.148497 0.988913i 0.452556π-0.452556\pi
0.148497 + 0.988913i 0.452556π0.452556\pi
422422 − 867.998i − 2.05687i
423423 0 0
424424 257.026 0.606194
425425 − 61.1584i − 0.143902i
426426 0 0
427427 −255.517 −0.598401
428428 712.093i 1.66377i
429429 0 0
430430 356.061 0.828049
431431 − 397.208i − 0.921596i −0.887505 0.460798i 0.847563π-0.847563\pi
0.887505 0.460798i 0.152437π-0.152437\pi
432432 0 0
433433 560.114 1.29357 0.646783 0.762674i 0.276114π-0.276114\pi
0.646783 + 0.762674i 0.276114π0.276114\pi
434434 1523.48i 3.51032i
435435 0 0
436436 121.956 0.279715
437437 76.1441i 0.174243i
438438 0 0
439439 −664.386 −1.51341 −0.756704 0.653758i 0.773191π-0.773191\pi
−0.756704 + 0.653758i 0.773191π0.773191\pi
440440 − 235.876i − 0.536081i
441441 0 0
442442 438.070 0.991108
443443 − 371.305i − 0.838160i −0.907949 0.419080i 0.862353π-0.862353\pi
0.907949 0.419080i 0.137647π-0.137647\pi
444444 0 0
445445 240.250 0.539887
446446 665.455i 1.49205i
447447 0 0
448448 −214.680 −0.479196
449449 585.471i 1.30395i 0.758243 + 0.651973i 0.226058π0.226058\pi
−0.758243 + 0.651973i 0.773942π0.773942\pi
450450 0 0
451451 −89.4121 −0.198253
452452 − 193.177i − 0.427382i
453453 0 0
454454 1482.30 3.26497
455455 − 157.133i − 0.345347i
456456 0 0
457457 168.641 0.369017 0.184508 0.982831i 0.440931π-0.440931\pi
0.184508 + 0.982831i 0.440931π0.440931\pi
458458 99.6462i 0.217568i
459459 0 0
460460 −251.026 −0.545709
461461 − 298.492i − 0.647487i −0.946145 0.323744i 0.895058π-0.895058\pi
0.946145 0.323744i 0.104942π-0.104942\pi
462462 0 0
463463 −595.285 −1.28571 −0.642856 0.765987i 0.722251π-0.722251\pi
−0.642856 + 0.765987i 0.722251π0.722251\pi
464464 1513.81i 3.26252i
465465 0 0
466466 −1300.89 −2.79162
467467 − 623.655i − 1.33545i −0.744408 0.667725i 0.767268π-0.767268\pi
0.744408 0.667725i 0.232732π-0.232732\pi
468468 0 0
469469 191.246 0.407773
470470 326.956i 0.695651i
471471 0 0
472472 488.855 1.03571
473473 236.756i 0.500542i
474474 0 0
475475 −31.6228 −0.0665743
476476 − 816.894i − 1.71616i
477477 0 0
478478 −991.745 −2.07478
479479 131.857i 0.275276i 0.990483 + 0.137638i 0.0439510π0.0439510\pi
−0.990483 + 0.137638i 0.956049π0.956049\pi
480480 0 0
481481 652.070 1.35565
482482 819.263i 1.69972i
483483 0 0
484484 −853.605 −1.76365
485485 2.41272i 0.00497468i
486486 0 0
487487 41.8028 0.0858375 0.0429187 0.999079i 0.486334π-0.486334\pi
0.0429187 + 0.999079i 0.486334π0.486334\pi
488488 − 693.392i − 1.42089i
489489 0 0
490490 −18.7587 −0.0382831
491491 178.817i 0.364189i 0.983281 + 0.182095i 0.0582877π0.0582877\pi
−0.983281 + 0.182095i 0.941712π0.941712\pi
492492 0 0
493493 −550.280 −1.11619
494494 − 226.510i − 0.458522i
495495 0 0
496496 −1960.80 −3.95322
497497 − 664.159i − 1.33634i
498498 0 0
499499 −39.0961 −0.0783489 −0.0391744 0.999232i 0.512473π-0.512473\pi
−0.0391744 + 0.999232i 0.512473π0.512473\pi
500500 − 104.252i − 0.208503i
501501 0 0
502502 −1163.62 −2.31796
503503 − 578.698i − 1.15049i −0.817980 0.575247i 0.804906π-0.804906\pi
0.817980 0.575247i 0.195094π-0.195094\pi
504504 0 0
505505 380.763 0.753986
506506 − 238.518i − 0.471379i
507507 0 0
508508 354.619 0.698068
509509 − 355.743i − 0.698905i −0.936954 0.349453i 0.886368π-0.886368\pi
0.936954 0.349453i 0.113632π-0.113632\pi
510510 0 0
511511 432.061 0.845521
512512 1098.99i 2.14646i
513513 0 0
514514 1357.81 2.64166
515515 287.246i 0.557760i
516516 0 0
517517 −217.404 −0.420510
518518 − 1737.57i − 3.35437i
519519 0 0
520520 426.408 0.820015
521521 810.952i 1.55653i 0.627936 + 0.778265i 0.283900π0.283900\pi
−0.627936 + 0.778265i 0.716100π0.716100\pi
522522 0 0
523523 720.483 1.37760 0.688798 0.724953i 0.258139π-0.258139\pi
0.688798 + 0.724953i 0.258139π0.258139\pi
524524 − 782.711i − 1.49372i
525525 0 0
526526 870.772 1.65546
527527 − 712.764i − 1.35249i
528528 0 0
529529 384.052 0.725996
530530 107.939i 0.203659i
531531 0 0
532532 −422.386 −0.793958
533533 − 161.636i − 0.303257i
534534 0 0
535535 −170.763 −0.319183
536536 518.979i 0.968245i
537537 0 0
538538 459.465 0.854024
539539 − 12.4733i − 0.0231415i
540540 0 0
541541 347.149 0.641680 0.320840 0.947133i 0.396035π-0.396035\pi
0.320840 + 0.947133i 0.396035π0.396035\pi
542542 944.142i 1.74196i
543543 0 0
544544 551.456 1.01371
545545 29.2455i 0.0536615i
546546 0 0
547547 −720.833 −1.31779 −0.658896 0.752234i 0.728976π-0.728976\pi
−0.658896 + 0.752234i 0.728976π0.728976\pi
548548 145.403i 0.265334i
549549 0 0
550550 99.0569 0.180104
551551 284.530i 0.516388i
552552 0 0
553553 −689.149 −1.24620
554554 − 831.222i − 1.50040i
555555 0 0
556556 −632.596 −1.13776
557557 429.102i 0.770380i 0.922837 + 0.385190i 0.125864π0.125864\pi
−0.922837 + 0.385190i 0.874136π0.874136\pi
558558 0 0
559559 −428.000 −0.765653
560560 − 538.902i − 0.962325i
561561 0 0
562562 −881.118 −1.56783
563563 670.820i 1.19151i 0.803166 + 0.595755i 0.203147π0.203147\pi
−0.803166 + 0.595755i 0.796853π0.796853\pi
564564 0 0
565565 46.3246 0.0819904
566566 − 760.474i − 1.34359i
567567 0 0
568568 1802.31 3.17309
569569 − 368.663i − 0.647914i −0.946072 0.323957i 0.894987π-0.894987\pi
0.946072 0.323957i 0.105013π-0.105013\pi
570570 0 0
571571 124.289 0.217669 0.108834 0.994060i 0.465288π-0.465288\pi
0.108834 + 0.994060i 0.465288π0.465288\pi
572572 496.532i 0.868063i
573573 0 0
574574 −430.710 −0.750366
575575 − 60.1972i − 0.104691i
576576 0 0
577577 504.236 0.873893 0.436947 0.899487i 0.356060π-0.356060\pi
0.436947 + 0.899487i 0.356060π0.356060\pi
578578 − 508.797i − 0.880272i
579579 0 0
580580 −938.017 −1.61727
581581 566.690i 0.975370i
582582 0 0
583583 −71.7722 −0.123108
584584 1172.48i 2.00766i
585585 0 0
586586 −736.236 −1.25638
587587 − 39.2256i − 0.0668238i −0.999442 0.0334119i 0.989363π-0.989363\pi
0.999442 0.0334119i 0.0106373π-0.0106373\pi
588588 0 0
589589 −368.544 −0.625711
590590 205.297i 0.347960i
591591 0 0
592592 2236.34 3.77760
593593 621.670i 1.04835i 0.851611 + 0.524174i 0.175626π0.175626\pi
−0.851611 + 0.524174i 0.824374π0.824374\pi
594594 0 0
595595 195.895 0.329235
596596 2176.59i 3.65200i
597597 0 0
598598 431.184 0.721044
599599 1119.77i 1.86940i 0.355436 + 0.934701i 0.384332π0.384332\pi
−0.355436 + 0.934701i 0.615668π0.615668\pi
600600 0 0
601601 −323.789 −0.538751 −0.269375 0.963035i 0.586817π-0.586817\pi
−0.269375 + 0.963035i 0.586817π0.586817\pi
602602 1140.49i 1.89450i
603603 0 0
604604 −1728.31 −2.86145
605605 − 204.698i − 0.338344i
606606 0 0
607607 1025.63 1.68966 0.844832 0.535031i 0.179700π-0.179700\pi
0.844832 + 0.535031i 0.179700π0.179700\pi
608608 − 285.138i − 0.468976i
609609 0 0
610610 291.193 0.477365
611611 − 393.014i − 0.643232i
612612 0 0
613613 904.153 1.47496 0.737482 0.675367i 0.236015π-0.236015\pi
0.737482 + 0.675367i 0.236015π0.236015\pi
614614 − 1251.40i − 2.03812i
615615 0 0
616616 755.526 1.22650
617617 − 710.716i − 1.15189i −0.817488 0.575945i 0.804634π-0.804634\pi
0.817488 0.575945i 0.195366π-0.195366\pi
618618 0 0
619619 −583.737 −0.943032 −0.471516 0.881858i 0.656293π-0.656293\pi
−0.471516 + 0.881858i 0.656293π0.656293\pi
620620 − 1214.99i − 1.95966i
621621 0 0
622622 −794.447 −1.27725
623623 769.537i 1.23521i
624624 0 0
625625 25.0000 0.0400000
626626 − 1028.99i − 1.64376i
627627 0 0
628628 1037.60 1.65223
629629 812.924i 1.29241i
630630 0 0
631631 −20.0968 −0.0318491 −0.0159246 0.999873i 0.505069π-0.505069\pi
−0.0159246 + 0.999873i 0.505069π0.505069\pi
632632 − 1870.13i − 2.95906i
633633 0 0
634634 56.2192 0.0886738
635635 85.0390i 0.133920i
636636 0 0
637637 22.5487 0.0353983
638638 − 891.277i − 1.39699i
639639 0 0
640640 −158.592 −0.247800
641641 341.607i 0.532928i 0.963845 + 0.266464i 0.0858553π0.0858553\pi
−0.963845 + 0.266464i 0.914145π0.914145\pi
642642 0 0
643643 −469.693 −0.730471 −0.365235 0.930915i 0.619011π-0.619011\pi
−0.365235 + 0.930915i 0.619011π0.619011\pi
644644 − 804.054i − 1.24853i
645645 0 0
646646 282.386 0.437130
647647 − 572.099i − 0.884234i −0.896957 0.442117i 0.854228π-0.854228\pi
0.896957 0.442117i 0.145772π-0.145772\pi
648648 0 0
649649 −136.508 −0.210336
650650 179.072i 0.275495i
651651 0 0
652652 −1107.41 −1.69848
653653 213.540i 0.327014i 0.986542 + 0.163507i 0.0522806π0.0522806\pi
−0.986542 + 0.163507i 0.947719π0.947719\pi
654654 0 0
655655 187.698 0.286561
656656 − 554.347i − 0.845040i
657657 0 0
658658 −1047.26 −1.59158
659659 − 420.983i − 0.638820i −0.947617 0.319410i 0.896515π-0.896515\pi
0.947617 0.319410i 0.103485π-0.103485\pi
660660 0 0
661661 434.272 0.656992 0.328496 0.944505i 0.393458π-0.393458\pi
0.328496 + 0.944505i 0.393458π0.393458\pi
662662 − 1370.74i − 2.07061i
663663 0 0
664664 −1537.81 −2.31599
665665 − 101.290i − 0.152316i
666666 0 0
667667 −541.631 −0.812041
668668 2062.54i 3.08763i
669669 0 0
670670 −217.947 −0.325295
671671 193.623i 0.288560i
672672 0 0
673673 72.7801 0.108143 0.0540714 0.998537i 0.482780π-0.482780\pi
0.0540714 + 0.998537i 0.482780π0.482780\pi
674674 − 686.669i − 1.01880i
675675 0 0
676676 678.237 1.00331
677677 172.106i 0.254219i 0.991889 + 0.127109i 0.0405700π0.0405700\pi
−0.991889 + 0.127109i 0.959430π0.959430\pi
678678 0 0
679679 −7.72811 −0.0113816
680680 531.595i 0.781758i
681681 0 0
682682 1154.45 1.69274
683683 − 792.592i − 1.16046i −0.814454 0.580228i 0.802963π-0.802963\pi
0.814454 0.580228i 0.197037π-0.197037\pi
684684 0 0
685685 −34.8683 −0.0509027
686686 1220.99i 1.77986i
687687 0 0
688688 −1467.87 −2.13353
689689 − 129.747i − 0.188313i
690690 0 0
691691 154.851 0.224097 0.112049 0.993703i 0.464259π-0.464259\pi
0.112049 + 0.993703i 0.464259π0.464259\pi
692692 1779.19i 2.57109i
693693 0 0
694694 1875.49 2.70244
695695 − 151.699i − 0.218272i
696696 0 0
697697 201.509 0.289109
698698 − 410.785i − 0.588518i
699699 0 0
700700 333.925 0.477036
701701 − 950.544i − 1.35598i −0.735070 0.677991i 0.762851π-0.762851\pi
0.735070 0.677991i 0.237149π-0.237149\pi
702702 0 0
703703 420.333 0.597913
704704 162.678i 0.231076i
705705 0 0
706706 −1562.95 −2.21381
707707 1219.61i 1.72505i
708708 0 0
709709 390.350 0.550564 0.275282 0.961363i 0.411229π-0.411229\pi
0.275282 + 0.961363i 0.411229π0.411229\pi
710710 756.889i 1.06604i
711711 0 0
712712 −2088.28 −2.93297
713713 − 701.561i − 0.983956i
714714 0 0
715715 −119.070 −0.166532
716716 − 541.762i − 0.756650i
717717 0 0
718718 −205.132 −0.285699
719719 655.227i 0.911303i 0.890158 + 0.455651i 0.150594π0.150594\pi
−0.890158 + 0.455651i 0.849406π0.849406\pi
720720 0 0
721721 −920.070 −1.27610
722722 1171.74i 1.62291i
723723 0 0
724724 1519.17 2.09830
725725 − 224.940i − 0.310263i
726726 0 0
727727 1424.25 1.95908 0.979539 0.201254i 0.0645017π-0.0645017\pi
0.979539 + 0.201254i 0.0645017π0.0645017\pi
728728 1365.81i 1.87612i
729729 0 0
730730 −492.386 −0.674501
731731 − 533.580i − 0.729932i
732732 0 0
733733 −946.749 −1.29161 −0.645805 0.763503i 0.723478π-0.723478\pi
−0.645805 + 0.763503i 0.723478π0.723478\pi
734734 563.471i 0.767672i
735735 0 0
736736 542.789 0.737485
737737 − 144.920i − 0.196635i
738738 0 0
739739 591.429 0.800310 0.400155 0.916447i 0.368956π-0.368956\pi
0.400155 + 0.916447i 0.368956π0.368956\pi
740740 1385.72i 1.87260i
741741 0 0
742742 −345.737 −0.465952
743743 − 732.202i − 0.985467i −0.870180 0.492734i 0.835998π-0.835998\pi
0.870180 0.492734i 0.164002π-0.164002\pi
744744 0 0
745745 −521.956 −0.700612
746746 − 2034.25i − 2.72687i
747747 0 0
748748 −619.018 −0.827564
749749 − 546.965i − 0.730261i
750750 0 0
751751 −215.359 −0.286764 −0.143382 0.989667i 0.545798π-0.545798\pi
−0.143382 + 0.989667i 0.545798π0.545798\pi
752752 − 1347.88i − 1.79240i
753753 0 0
754754 1611.22 2.13689
755755 − 414.457i − 0.548950i
756756 0 0
757757 276.258 0.364938 0.182469 0.983212i 0.441591π-0.441591\pi
0.182469 + 0.983212i 0.441591π0.441591\pi
758758 − 538.064i − 0.709848i
759759 0 0
760760 274.868 0.361669
761761 − 893.373i − 1.17395i −0.809606 0.586973i 0.800319π-0.800319\pi
0.809606 0.586973i 0.199681π-0.199681\pi
762762 0 0
763763 −93.6754 −0.122773
764764 933.033i 1.22125i
765765 0 0
766766 2688.87 3.51027
767767 − 246.775i − 0.321740i
768768 0 0
769769 284.316 0.369722 0.184861 0.982765i 0.440817π-0.440817\pi
0.184861 + 0.982765i 0.440817π0.440817\pi
770770 317.286i 0.412060i
771771 0 0
772772 577.140 0.747591
773773 1059.64i 1.37081i 0.728162 + 0.685405i 0.240375π0.240375\pi
−0.728162 + 0.685405i 0.759625π0.759625\pi
774774 0 0
775775 291.359 0.375948
776776 − 20.9716i − 0.0270253i
777777 0 0
778778 1081.97 1.39071
779779 − 104.193i − 0.133752i
780780 0 0
781781 −503.280 −0.644404
782782 537.550i 0.687404i
783783 0 0
784784 77.3331 0.0986392
785785 248.821i 0.316970i
786786 0 0
787787 −875.517 −1.11247 −0.556237 0.831024i 0.687755π-0.687755\pi
−0.556237 + 0.831024i 0.687755π0.687755\pi
788788 − 231.986i − 0.294399i
789789 0 0
790790 785.368 0.994137
791791 148.381i 0.187586i
792792 0 0
793793 −350.026 −0.441394
794794 1668.39i 2.10124i
795795 0 0
796796 1459.21 1.83318
797797 − 742.449i − 0.931555i −0.884902 0.465777i 0.845775π-0.845775\pi
0.884902 0.465777i 0.154225π-0.154225\pi
798798 0 0
799799 489.964 0.613222
800800 225.421i 0.281776i
801801 0 0
802802 1427.78 1.78027
803803 − 327.403i − 0.407725i
804804 0 0
805805 192.816 0.239523
806806 2086.97i 2.58929i
807807 0 0
808808 −3309.63 −4.09608
809809 − 113.720i − 0.140568i −0.997527 0.0702842i 0.977609π-0.977609\pi
0.997527 0.0702842i 0.0223906π-0.0223906\pi
810810 0 0
811811 −1466.03 −1.80769 −0.903844 0.427863i 0.859267π-0.859267\pi
−0.903844 + 0.427863i 0.859267π0.859267\pi
812812 − 3004.53i − 3.70016i
813813 0 0
814814 −1316.67 −1.61754
815815 − 265.562i − 0.325843i
816816 0 0
817817 −275.895 −0.337692
818818 1503.34i 1.83782i
819819 0 0
820820 343.495 0.418897
821821 − 550.073i − 0.670003i −0.942218 0.335002i 0.891263π-0.891263\pi
0.942218 0.335002i 0.108737π-0.108737\pi
822822 0 0
823823 1392.51 1.69199 0.845997 0.533187i 0.179006π-0.179006\pi
0.845997 + 0.533187i 0.179006π0.179006\pi
824824 − 2496.77i − 3.03007i
825825 0 0
826826 −657.579 −0.796100
827827 955.922i 1.15589i 0.816075 + 0.577945i 0.196145π0.196145\pi
−0.816075 + 0.577945i 0.803855π0.803855\pi
828828 0 0
829829 −1652.69 −1.99360 −0.996798 0.0799552i 0.974522π-0.974522\pi
−0.996798 + 0.0799552i 0.974522π0.974522\pi
830830 − 645.811i − 0.778086i
831831 0 0
832832 −294.083 −0.353465
833833 28.1111i 0.0337469i
834834 0 0
835835 −494.605 −0.592341
836836 320.071i 0.382860i
837837 0 0
838838 −2385.27 −2.84638
839839 1568.11i 1.86903i 0.355927 + 0.934514i 0.384165π0.384165\pi
−0.355927 + 0.934514i 0.615835π0.615835\pi
840840 0 0
841841 −1182.93 −1.40657
842842 − 456.413i − 0.542058i
843843 0 0
844844 −2217.28 −2.62711
845845 162.644i 0.192478i
846846 0 0
847847 655.662 0.774099
848848 − 444.981i − 0.524742i
849849 0 0
850850 −223.246 −0.262642
851851 800.147i 0.940243i
852852 0 0
853853 −651.232 −0.763461 −0.381730 0.924274i 0.624672π-0.624672\pi
−0.381730 + 0.924274i 0.624672π0.624672\pi
854854 932.711i 1.09217i
855855 0 0
856856 1484.29 1.73398
857857 299.131i 0.349044i 0.984653 + 0.174522i 0.0558380π0.0558380\pi
−0.984653 + 0.174522i 0.944162π0.944162\pi
858858 0 0
859859 1095.72 1.27558 0.637788 0.770212i 0.279850π-0.279850\pi
0.637788 + 0.770212i 0.279850π0.279850\pi
860860 − 909.550i − 1.05762i
861861 0 0
862862 −1449.92 −1.68204
863863 1221.95i 1.41593i 0.706247 + 0.707965i 0.250387π0.250387\pi
−0.706247 + 0.707965i 0.749613π0.749613\pi
864864 0 0
865865 −426.658 −0.493246
866866 − 2044.57i − 2.36094i
867867 0 0
868868 3891.69 4.48352
869869 522.216i 0.600939i
870870 0 0
871871 261.982 0.300782
872872 − 254.205i − 0.291520i
873873 0 0
874874 277.947 0.318018
875875 80.0767i 0.0915162i
876876 0 0
877877 766.399 0.873887 0.436944 0.899489i 0.356061π-0.356061\pi
0.436944 + 0.899489i 0.356061π0.356061\pi
878878 2425.20i 2.76218i
879879 0 0
880880 −408.364 −0.464050
881881 − 310.097i − 0.351983i −0.984392 0.175992i 0.943687π-0.943687\pi
0.984392 0.175992i 0.0563132π-0.0563132\pi
882882 0 0
883883 −122.236 −0.138433 −0.0692165 0.997602i 0.522050π-0.522050\pi
−0.0692165 + 0.997602i 0.522050π0.522050\pi
884884 − 1119.04i − 1.26588i
885885 0 0
886886 −1355.37 −1.52976
887887 265.444i 0.299261i 0.988742 + 0.149630i 0.0478084π0.0478084\pi
−0.988742 + 0.149630i 0.952192π0.952192\pi
888888 0 0
889889 −272.386 −0.306396
890890 − 876.980i − 0.985370i
891891 0 0
892892 1699.89 1.90571
893893 − 253.343i − 0.283698i
894894 0 0
895895 129.917 0.145158
896896 − 507.981i − 0.566944i
897897 0 0
898898 2137.14 2.37988
899899 − 2621.54i − 2.91606i
900900 0 0
901901 161.754 0.179527
902902 326.379i 0.361840i
903903 0 0
904904 −402.658 −0.445418
905905 364.302i 0.402544i
906906 0 0
907907 672.622 0.741590 0.370795 0.928715i 0.379085π-0.379085\pi
0.370795 + 0.928715i 0.379085π0.379085\pi
908908 − 3786.49i − 4.17015i
909909 0 0
910910 −573.579 −0.630306
911911 1402.48i 1.53949i 0.638350 + 0.769746i 0.279617π0.279617\pi
−0.638350 + 0.769746i 0.720383π0.720383\pi
912912 0 0
913913 429.421 0.470340
914914 − 615.586i − 0.673507i
915915 0 0
916916 254.544 0.277886
917917 601.208i 0.655625i
918918 0 0
919919 −338.255 −0.368068 −0.184034 0.982920i 0.558916π-0.558916\pi
−0.184034 + 0.982920i 0.558916π0.558916\pi
920920 523.240i 0.568739i
921921 0 0
922922 −1089.58 −1.18176
923923 − 909.811i − 0.985711i
924924 0 0
925925 −332.302 −0.359246
926926 2172.96i 2.34661i
927927 0 0
928928 2028.25 2.18562
929929 148.207i 0.159533i 0.996814 + 0.0797667i 0.0254175π0.0254175\pi
−0.996814 + 0.0797667i 0.974582π0.974582\pi
930930 0 0
931931 14.5352 0.0156125
932932 3323.10i 3.56556i
933933 0 0
934934 −2276.52 −2.43738
935935 − 148.443i − 0.158763i
936936 0 0
937937 −1416.72 −1.51197 −0.755987 0.654587i 0.772843π-0.772843\pi
−0.755987 + 0.654587i 0.772843π0.772843\pi
938938 − 698.100i − 0.744243i
939939 0 0
940940 835.201 0.888512
941941 − 1398.92i − 1.48663i −0.668939 0.743317i 0.733251π-0.733251\pi
0.668939 0.743317i 0.266749π-0.266749\pi
942942 0 0
943943 198.342 0.210330
944944 − 846.338i − 0.896544i
945945 0 0
946946 864.228 0.913560
947947 1050.57i 1.10937i 0.832060 + 0.554686i 0.187161π0.187161\pi
−0.832060 + 0.554686i 0.812839π0.812839\pi
948948 0 0
949949 591.868 0.623675
950950 115.432i 0.121507i
951951 0 0
952952 −1702.74 −1.78859
953953 − 551.928i − 0.579148i −0.957156 0.289574i 0.906486π-0.906486\pi
0.957156 0.289574i 0.0935136π-0.0935136\pi
954954 0 0
955955 −223.745 −0.234288
956956 2533.39i 2.64999i
957957 0 0
958958 481.315 0.502417
959959 − 111.686i − 0.116461i
960960 0 0
961961 2434.61 2.53342
962962 − 2380.24i − 2.47426i
963963 0 0
964964 2092.79 2.17094
965965 138.401i 0.143420i
966966 0 0
967967 −357.093 −0.369279 −0.184639 0.982806i 0.559112π-0.559112\pi
−0.184639 + 0.982806i 0.559112π0.559112\pi
968968 1779.26i 1.83807i
969969 0 0
970970 8.80711 0.00907950
971971 308.206i 0.317411i 0.987326 + 0.158705i 0.0507320π0.0507320\pi
−0.987326 + 0.158705i 0.949268π0.949268\pi
972972 0 0
973973 485.903 0.499387
974974 − 152.592i − 0.156665i
975975 0 0
976976 −1200.45 −1.22997
977977 253.280i 0.259243i 0.991564 + 0.129621i 0.0413762π0.0413762\pi
−0.991564 + 0.129621i 0.958624π0.958624\pi
978978 0 0
979979 583.132 0.595640
980980 47.9187i 0.0488966i
981981 0 0
982982 652.732 0.664697
983983 1068.73i 1.08722i 0.839339 + 0.543609i 0.182942π0.182942\pi
−0.839339 + 0.543609i 0.817058π0.817058\pi
984984 0 0
985985 55.6313 0.0564785
986986 2008.68i 2.03720i
987987 0 0
988988 −578.614 −0.585641
989989 − 525.194i − 0.531035i
990990 0 0
991991 280.631 0.283179 0.141590 0.989925i 0.454779π-0.454779\pi
0.141590 + 0.989925i 0.454779π0.454779\pi
992992 2627.14i 2.64833i
993993 0 0
994994 −2424.37 −2.43900
995995 349.925i 0.351683i
996996 0 0
997997 −356.574 −0.357647 −0.178824 0.983881i 0.557229π-0.557229\pi
−0.178824 + 0.983881i 0.557229π0.557229\pi
998998 142.712i 0.142998i
999999 0 0
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 45.3.c.a.26.1 4
3.2 odd 2 inner 45.3.c.a.26.4 yes 4
4.3 odd 2 720.3.l.a.161.1 4
5.2 odd 4 225.3.d.b.224.8 8
5.3 odd 4 225.3.d.b.224.1 8
5.4 even 2 225.3.c.c.26.4 4
8.3 odd 2 2880.3.l.c.1601.3 4
8.5 even 2 2880.3.l.g.1601.4 4
9.2 odd 6 405.3.i.d.296.4 8
9.4 even 3 405.3.i.d.26.4 8
9.5 odd 6 405.3.i.d.26.1 8
9.7 even 3 405.3.i.d.296.1 8
12.11 even 2 720.3.l.a.161.3 4
15.2 even 4 225.3.d.b.224.2 8
15.8 even 4 225.3.d.b.224.7 8
15.14 odd 2 225.3.c.c.26.1 4
20.3 even 4 3600.3.c.i.449.8 8
20.7 even 4 3600.3.c.i.449.2 8
20.19 odd 2 3600.3.l.v.1601.4 4
24.5 odd 2 2880.3.l.g.1601.2 4
24.11 even 2 2880.3.l.c.1601.1 4
60.23 odd 4 3600.3.c.i.449.7 8
60.47 odd 4 3600.3.c.i.449.1 8
60.59 even 2 3600.3.l.v.1601.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.3.c.a.26.1 4 1.1 even 1 trivial
45.3.c.a.26.4 yes 4 3.2 odd 2 inner
225.3.c.c.26.1 4 15.14 odd 2
225.3.c.c.26.4 4 5.4 even 2
225.3.d.b.224.1 8 5.3 odd 4
225.3.d.b.224.2 8 15.2 even 4
225.3.d.b.224.7 8 15.8 even 4
225.3.d.b.224.8 8 5.2 odd 4
405.3.i.d.26.1 8 9.5 odd 6
405.3.i.d.26.4 8 9.4 even 3
405.3.i.d.296.1 8 9.7 even 3
405.3.i.d.296.4 8 9.2 odd 6
720.3.l.a.161.1 4 4.3 odd 2
720.3.l.a.161.3 4 12.11 even 2
2880.3.l.c.1601.1 4 24.11 even 2
2880.3.l.c.1601.3 4 8.3 odd 2
2880.3.l.g.1601.2 4 24.5 odd 2
2880.3.l.g.1601.4 4 8.5 even 2
3600.3.c.i.449.1 8 60.47 odd 4
3600.3.c.i.449.2 8 20.7 even 4
3600.3.c.i.449.7 8 60.23 odd 4
3600.3.c.i.449.8 8 20.3 even 4
3600.3.l.v.1601.3 4 60.59 even 2
3600.3.l.v.1601.4 4 20.19 odd 2