Properties

Label 50.4.b.a.49.2
Level 5050
Weight 44
Character 50.49
Analytic conductor 2.9502.950
Analytic rank 00
Dimension 22
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,4,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 50=252 50 = 2 \cdot 5^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 50.b (of order 22, degree 11, not 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: 2.950095500292.95009550029
Analytic rank: 00
Dimension: 22
Coefficient field: Q(i)\Q(i)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x2+1 x^{2} + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 2 2
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 49.2
Root 1.00000i1.00000i of defining polynomial
Character χ\chi == 50.49
Dual form 50.4.b.a.49.1

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) == q+2.00000iq2+8.00000iq34.00000q416.0000q64.00000iq78.00000iq837.0000q9+12.0000q1132.0000iq12+58.0000iq13+8.00000q14+16.0000q16+66.0000iq1774.0000iq18+100.000q19+32.0000q21+24.0000iq22132.000iq23+64.0000q24116.000q2680.0000iq27+16.0000iq28+90.0000q29+152.000q31+32.0000iq32+96.0000iq33132.000q34+148.000q3634.0000iq37+200.000iq38464.000q39438.000q41+64.0000iq4232.0000iq4348.0000q44+264.000q46204.000iq47+128.000iq48+327.000q49528.000q51232.000iq52222.000iq53+160.000q5432.0000q56+800.000iq57+180.000iq58420.000q59+902.000q61+304.000iq62+148.000iq6364.0000q64192.000q661024.00iq67264.000iq68+1056.00q69+432.000q71+296.000iq72362.000iq73+68.0000q74400.000q7648.0000iq77928.000iq78+160.000q79359.000q81876.000iq8272.0000iq83128.000q84+64.0000q86+720.000iq8796.0000iq88810.000q89+232.000q91+528.000iq92+1216.00iq93+408.000q94256.000q96+1106.00iq97+654.000iq98444.000q99+O(q100)q+2.00000i q^{2} +8.00000i q^{3} -4.00000 q^{4} -16.0000 q^{6} -4.00000i q^{7} -8.00000i q^{8} -37.0000 q^{9} +12.0000 q^{11} -32.0000i q^{12} +58.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} +66.0000i q^{17} -74.0000i q^{18} +100.000 q^{19} +32.0000 q^{21} +24.0000i q^{22} -132.000i q^{23} +64.0000 q^{24} -116.000 q^{26} -80.0000i q^{27} +16.0000i q^{28} +90.0000 q^{29} +152.000 q^{31} +32.0000i q^{32} +96.0000i q^{33} -132.000 q^{34} +148.000 q^{36} -34.0000i q^{37} +200.000i q^{38} -464.000 q^{39} -438.000 q^{41} +64.0000i q^{42} -32.0000i q^{43} -48.0000 q^{44} +264.000 q^{46} -204.000i q^{47} +128.000i q^{48} +327.000 q^{49} -528.000 q^{51} -232.000i q^{52} -222.000i q^{53} +160.000 q^{54} -32.0000 q^{56} +800.000i q^{57} +180.000i q^{58} -420.000 q^{59} +902.000 q^{61} +304.000i q^{62} +148.000i q^{63} -64.0000 q^{64} -192.000 q^{66} -1024.00i q^{67} -264.000i q^{68} +1056.00 q^{69} +432.000 q^{71} +296.000i q^{72} -362.000i q^{73} +68.0000 q^{74} -400.000 q^{76} -48.0000i q^{77} -928.000i q^{78} +160.000 q^{79} -359.000 q^{81} -876.000i q^{82} -72.0000i q^{83} -128.000 q^{84} +64.0000 q^{86} +720.000i q^{87} -96.0000i q^{88} -810.000 q^{89} +232.000 q^{91} +528.000i q^{92} +1216.00i q^{93} +408.000 q^{94} -256.000 q^{96} +1106.00i q^{97} +654.000i q^{98} -444.000 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q8q432q674q9+24q11+16q14+32q16+200q19+64q21+128q24232q26+180q29+304q31264q34+296q36928q39876q4196q44+888q99+O(q100) 2 q - 8 q^{4} - 32 q^{6} - 74 q^{9} + 24 q^{11} + 16 q^{14} + 32 q^{16} + 200 q^{19} + 64 q^{21} + 128 q^{24} - 232 q^{26} + 180 q^{29} + 304 q^{31} - 264 q^{34} + 296 q^{36} - 928 q^{39} - 876 q^{41} - 96 q^{44}+ \cdots - 888 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/50Z)×\left(\mathbb{Z}/50\mathbb{Z}\right)^\times.

nn 2727
χ(n)\chi(n) 1-1

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 2.00000i 0.707107i
33 8.00000i 1.53960i 0.638285 + 0.769800i 0.279644π0.279644\pi
−0.638285 + 0.769800i 0.720356π0.720356\pi
44 −4.00000 −0.500000
55 0 0
66 −16.0000 −1.08866
77 − 4.00000i − 0.215980i −0.994152 0.107990i 0.965559π-0.965559\pi
0.994152 0.107990i 0.0344414π-0.0344414\pi
88 − 8.00000i − 0.353553i
99 −37.0000 −1.37037
1010 0 0
1111 12.0000 0.328921 0.164461 0.986384i 0.447412π-0.447412\pi
0.164461 + 0.986384i 0.447412π0.447412\pi
1212 − 32.0000i − 0.769800i
1313 58.0000i 1.23741i 0.785624 + 0.618704i 0.212342π0.212342\pi
−0.785624 + 0.618704i 0.787658π0.787658\pi
1414 8.00000 0.152721
1515 0 0
1616 16.0000 0.250000
1717 66.0000i 0.941609i 0.882238 + 0.470804i 0.156036π0.156036\pi
−0.882238 + 0.470804i 0.843964π0.843964\pi
1818 − 74.0000i − 0.968998i
1919 100.000 1.20745 0.603726 0.797192i 0.293682π-0.293682\pi
0.603726 + 0.797192i 0.293682π0.293682\pi
2020 0 0
2121 32.0000 0.332522
2222 24.0000i 0.232583i
2323 − 132.000i − 1.19669i −0.801238 0.598346i 0.795825π-0.795825\pi
0.801238 0.598346i 0.204175π-0.204175\pi
2424 64.0000 0.544331
2525 0 0
2626 −116.000 −0.874980
2727 − 80.0000i − 0.570222i
2828 16.0000i 0.107990i
2929 90.0000 0.576296 0.288148 0.957586i 0.406961π-0.406961\pi
0.288148 + 0.957586i 0.406961π0.406961\pi
3030 0 0
3131 152.000 0.880645 0.440323 0.897840i 0.354864π-0.354864\pi
0.440323 + 0.897840i 0.354864π0.354864\pi
3232 32.0000i 0.176777i
3333 96.0000i 0.506408i
3434 −132.000 −0.665818
3535 0 0
3636 148.000 0.685185
3737 − 34.0000i − 0.151069i −0.997143 0.0755347i 0.975934π-0.975934\pi
0.997143 0.0755347i 0.0240664π-0.0240664\pi
3838 200.000i 0.853797i
3939 −464.000 −1.90511
4040 0 0
4141 −438.000 −1.66839 −0.834196 0.551467i 0.814068π-0.814068\pi
−0.834196 + 0.551467i 0.814068π0.814068\pi
4242 64.0000i 0.235129i
4343 − 32.0000i − 0.113487i −0.998389 0.0567437i 0.981928π-0.981928\pi
0.998389 0.0567437i 0.0180718π-0.0180718\pi
4444 −48.0000 −0.164461
4545 0 0
4646 264.000 0.846189
4747 − 204.000i − 0.633116i −0.948573 0.316558i 0.897473π-0.897473\pi
0.948573 0.316558i 0.102527π-0.102527\pi
4848 128.000i 0.384900i
4949 327.000 0.953353
5050 0 0
5151 −528.000 −1.44970
5252 − 232.000i − 0.618704i
5353 − 222.000i − 0.575359i −0.957727 0.287680i 0.907116π-0.907116\pi
0.957727 0.287680i 0.0928838π-0.0928838\pi
5454 160.000 0.403208
5555 0 0
5656 −32.0000 −0.0763604
5757 800.000i 1.85899i
5858 180.000i 0.407503i
5959 −420.000 −0.926769 −0.463384 0.886157i 0.653365π-0.653365\pi
−0.463384 + 0.886157i 0.653365π0.653365\pi
6060 0 0
6161 902.000 1.89327 0.946633 0.322312i 0.104460π-0.104460\pi
0.946633 + 0.322312i 0.104460π0.104460\pi
6262 304.000i 0.622710i
6363 148.000i 0.295972i
6464 −64.0000 −0.125000
6565 0 0
6666 −192.000 −0.358084
6767 − 1024.00i − 1.86719i −0.358334 0.933593i 0.616655π-0.616655\pi
0.358334 0.933593i 0.383345π-0.383345\pi
6868 − 264.000i − 0.470804i
6969 1056.00 1.84243
7070 0 0
7171 432.000 0.722098 0.361049 0.932547i 0.382419π-0.382419\pi
0.361049 + 0.932547i 0.382419π0.382419\pi
7272 296.000i 0.484499i
7373 − 362.000i − 0.580396i −0.956967 0.290198i 0.906279π-0.906279\pi
0.956967 0.290198i 0.0937211π-0.0937211\pi
7474 68.0000 0.106822
7575 0 0
7676 −400.000 −0.603726
7777 − 48.0000i − 0.0710404i
7878 − 928.000i − 1.34712i
7979 160.000 0.227866 0.113933 0.993488i 0.463655π-0.463655\pi
0.113933 + 0.993488i 0.463655π0.463655\pi
8080 0 0
8181 −359.000 −0.492455
8282 − 876.000i − 1.17973i
8383 − 72.0000i − 0.0952172i −0.998866 0.0476086i 0.984840π-0.984840\pi
0.998866 0.0476086i 0.0151600π-0.0151600\pi
8484 −128.000 −0.166261
8585 0 0
8686 64.0000 0.0802476
8787 720.000i 0.887266i
8888 − 96.0000i − 0.116291i
8989 −810.000 −0.964717 −0.482359 0.875974i 0.660220π-0.660220\pi
−0.482359 + 0.875974i 0.660220π0.660220\pi
9090 0 0
9191 232.000 0.267255
9292 528.000i 0.598346i
9393 1216.00i 1.35584i
9494 408.000 0.447681
9595 0 0
9696 −256.000 −0.272166
9797 1106.00i 1.15770i 0.815433 + 0.578852i 0.196499π0.196499\pi
−0.815433 + 0.578852i 0.803501π0.803501\pi
9898 654.000i 0.674122i
9999 −444.000 −0.450744
100100 0 0
101101 −258.000 −0.254178 −0.127089 0.991891i 0.540563π-0.540563\pi
−0.127089 + 0.991891i 0.540563π0.540563\pi
102102 − 1056.00i − 1.02509i
103103 988.000i 0.945151i 0.881290 + 0.472575i 0.156676π0.156676\pi
−0.881290 + 0.472575i 0.843324π0.843324\pi
104104 464.000 0.437490
105105 0 0
106106 444.000 0.406840
107107 − 24.0000i − 0.0216838i −0.999941 0.0108419i 0.996549π-0.996549\pi
0.999941 0.0108419i 0.00345115π-0.00345115\pi
108108 320.000i 0.285111i
109109 −950.000 −0.834803 −0.417401 0.908722i 0.637059π-0.637059\pi
−0.417401 + 0.908722i 0.637059π0.637059\pi
110110 0 0
111111 272.000 0.232586
112112 − 64.0000i − 0.0539949i
113113 1038.00i 0.864131i 0.901842 + 0.432066i 0.142215π0.142215\pi
−0.901842 + 0.432066i 0.857785π0.857785\pi
114114 −1600.00 −1.31451
115115 0 0
116116 −360.000 −0.288148
117117 − 2146.00i − 1.69571i
118118 − 840.000i − 0.655324i
119119 264.000 0.203368
120120 0 0
121121 −1187.00 −0.891811
122122 1804.00i 1.33874i
123123 − 3504.00i − 2.56866i
124124 −608.000 −0.440323
125125 0 0
126126 −296.000 −0.209284
127127 − 124.000i − 0.0866395i −0.999061 0.0433198i 0.986207π-0.986207\pi
0.999061 0.0433198i 0.0137934π-0.0137934\pi
128128 − 128.000i − 0.0883883i
129129 256.000 0.174725
130130 0 0
131131 132.000 0.0880374 0.0440187 0.999031i 0.485984π-0.485984\pi
0.0440187 + 0.999031i 0.485984π0.485984\pi
132132 − 384.000i − 0.253204i
133133 − 400.000i − 0.260785i
134134 2048.00 1.32030
135135 0 0
136136 528.000 0.332909
137137 − 1254.00i − 0.782018i −0.920387 0.391009i 0.872126π-0.872126\pi
0.920387 0.391009i 0.127874π-0.127874\pi
138138 2112.00i 1.30279i
139139 2860.00 1.74519 0.872597 0.488440i 0.162434π-0.162434\pi
0.872597 + 0.488440i 0.162434π0.162434\pi
140140 0 0
141141 1632.00 0.974746
142142 864.000i 0.510600i
143143 696.000i 0.407010i
144144 −592.000 −0.342593
145145 0 0
146146 724.000 0.410402
147147 2616.00i 1.46778i
148148 136.000i 0.0755347i
149149 −750.000 −0.412365 −0.206183 0.978514i 0.566104π-0.566104\pi
−0.206183 + 0.978514i 0.566104π0.566104\pi
150150 0 0
151151 −448.000 −0.241442 −0.120721 0.992686i 0.538521π-0.538521\pi
−0.120721 + 0.992686i 0.538521π0.538521\pi
152152 − 800.000i − 0.426898i
153153 − 2442.00i − 1.29035i
154154 96.0000 0.0502331
155155 0 0
156156 1856.00 0.952557
157157 2246.00i 1.14172i 0.821047 + 0.570861i 0.193390π0.193390\pi
−0.821047 + 0.570861i 0.806610π0.806610\pi
158158 320.000i 0.161126i
159159 1776.00 0.885824
160160 0 0
161161 −528.000 −0.258461
162162 − 718.000i − 0.348219i
163163 568.000i 0.272940i 0.990644 + 0.136470i 0.0435757π0.0435757\pi
−0.990644 + 0.136470i 0.956424π0.956424\pi
164164 1752.00 0.834196
165165 0 0
166166 144.000 0.0673287
167167 − 1524.00i − 0.706172i −0.935591 0.353086i 0.885132π-0.885132\pi
0.935591 0.353086i 0.114868π-0.114868\pi
168168 − 256.000i − 0.117564i
169169 −1167.00 −0.531179
170170 0 0
171171 −3700.00 −1.65466
172172 128.000i 0.0567437i
173173 − 3702.00i − 1.62692i −0.581618 0.813462i 0.697580π-0.697580\pi
0.581618 0.813462i 0.302420π-0.302420\pi
174174 −1440.00 −0.627391
175175 0 0
176176 192.000 0.0822304
177177 − 3360.00i − 1.42685i
178178 − 1620.00i − 0.682158i
179179 −3180.00 −1.32785 −0.663923 0.747801i 0.731110π-0.731110\pi
−0.663923 + 0.747801i 0.731110π0.731110\pi
180180 0 0
181181 −2098.00 −0.861564 −0.430782 0.902456i 0.641762π-0.641762\pi
−0.430782 + 0.902456i 0.641762π0.641762\pi
182182 464.000i 0.188978i
183183 7216.00i 2.91487i
184184 −1056.00 −0.423094
185185 0 0
186186 −2432.00 −0.958725
187187 792.000i 0.309715i
188188 816.000i 0.316558i
189189 −320.000 −0.123156
190190 0 0
191191 4392.00 1.66384 0.831921 0.554894i 0.187241π-0.187241\pi
0.831921 + 0.554894i 0.187241π0.187241\pi
192192 − 512.000i − 0.192450i
193193 2158.00i 0.804851i 0.915453 + 0.402425i 0.131833π0.131833\pi
−0.915453 + 0.402425i 0.868167π0.868167\pi
194194 −2212.00 −0.818620
195195 0 0
196196 −1308.00 −0.476676
197197 − 1074.00i − 0.388423i −0.980960 0.194212i 0.937785π-0.937785\pi
0.980960 0.194212i 0.0622148π-0.0622148\pi
198198 − 888.000i − 0.318724i
199199 −2840.00 −1.01167 −0.505835 0.862630i 0.668815π-0.668815\pi
−0.505835 + 0.862630i 0.668815π0.668815\pi
200200 0 0
201201 8192.00 2.87472
202202 − 516.000i − 0.179731i
203203 − 360.000i − 0.124468i
204204 2112.00 0.724851
205205 0 0
206206 −1976.00 −0.668323
207207 4884.00i 1.63991i
208208 928.000i 0.309352i
209209 1200.00 0.397157
210210 0 0
211211 −2668.00 −0.870487 −0.435243 0.900313i 0.643338π-0.643338\pi
−0.435243 + 0.900313i 0.643338π0.643338\pi
212212 888.000i 0.287680i
213213 3456.00i 1.11174i
214214 48.0000 0.0153328
215215 0 0
216216 −640.000 −0.201604
217217 − 608.000i − 0.190202i
218218 − 1900.00i − 0.590295i
219219 2896.00 0.893578
220220 0 0
221221 −3828.00 −1.16515
222222 544.000i 0.164463i
223223 − 1772.00i − 0.532116i −0.963957 0.266058i 0.914279π-0.914279\pi
0.963957 0.266058i 0.0857213π-0.0857213\pi
224224 128.000 0.0381802
225225 0 0
226226 −2076.00 −0.611033
227227 − 2784.00i − 0.814011i −0.913426 0.407006i 0.866573π-0.866573\pi
0.913426 0.407006i 0.133427π-0.133427\pi
228228 − 3200.00i − 0.929496i
229229 −350.000 −0.100998 −0.0504992 0.998724i 0.516081π-0.516081\pi
−0.0504992 + 0.998724i 0.516081π0.516081\pi
230230 0 0
231231 384.000 0.109374
232232 − 720.000i − 0.203751i
233233 − 1962.00i − 0.551652i −0.961208 0.275826i 0.911049π-0.911049\pi
0.961208 0.275826i 0.0889513π-0.0889513\pi
234234 4292.00 1.19905
235235 0 0
236236 1680.00 0.463384
237237 1280.00i 0.350823i
238238 528.000i 0.143803i
239239 4320.00 1.16919 0.584597 0.811324i 0.301252π-0.301252\pi
0.584597 + 0.811324i 0.301252π0.301252\pi
240240 0 0
241241 −478.000 −0.127762 −0.0638811 0.997958i 0.520348π-0.520348\pi
−0.0638811 + 0.997958i 0.520348π0.520348\pi
242242 − 2374.00i − 0.630605i
243243 − 5032.00i − 1.32841i
244244 −3608.00 −0.946633
245245 0 0
246246 7008.00 1.81632
247247 5800.00i 1.49411i
248248 − 1216.00i − 0.311355i
249249 576.000 0.146596
250250 0 0
251251 2652.00 0.666903 0.333452 0.942767i 0.391787π-0.391787\pi
0.333452 + 0.942767i 0.391787π0.391787\pi
252252 − 592.000i − 0.147986i
253253 − 1584.00i − 0.393617i
254254 248.000 0.0612634
255255 0 0
256256 256.000 0.0625000
257257 − 2334.00i − 0.566502i −0.959046 0.283251i 0.908587π-0.908587\pi
0.959046 0.283251i 0.0914129π-0.0914129\pi
258258 512.000i 0.123549i
259259 −136.000 −0.0326279
260260 0 0
261261 −3330.00 −0.789739
262262 264.000i 0.0622518i
263263 3948.00i 0.925643i 0.886451 + 0.462822i 0.153163π0.153163\pi
−0.886451 + 0.462822i 0.846837π0.846837\pi
264264 768.000 0.179042
265265 0 0
266266 800.000 0.184403
267267 − 6480.00i − 1.48528i
268268 4096.00i 0.933593i
269269 −1590.00 −0.360387 −0.180193 0.983631i 0.557672π-0.557672\pi
−0.180193 + 0.983631i 0.557672π0.557672\pi
270270 0 0
271271 4952.00 1.11001 0.555005 0.831847i 0.312716π-0.312716\pi
0.555005 + 0.831847i 0.312716π0.312716\pi
272272 1056.00i 0.235402i
273273 1856.00i 0.411466i
274274 2508.00 0.552970
275275 0 0
276276 −4224.00 −0.921213
277277 1646.00i 0.357034i 0.983937 + 0.178517i 0.0571300π0.0571300\pi
−0.983937 + 0.178517i 0.942870π0.942870\pi
278278 5720.00i 1.23404i
279279 −5624.00 −1.20681
280280 0 0
281281 −1158.00 −0.245838 −0.122919 0.992417i 0.539226π-0.539226\pi
−0.122919 + 0.992417i 0.539226π0.539226\pi
282282 3264.00i 0.689250i
283283 − 6992.00i − 1.46866i −0.678792 0.734331i 0.737496π-0.737496\pi
0.678792 0.734331i 0.262504π-0.262504\pi
284284 −1728.00 −0.361049
285285 0 0
286286 −1392.00 −0.287800
287287 1752.00i 0.360339i
288288 − 1184.00i − 0.242250i
289289 557.000 0.113373
290290 0 0
291291 −8848.00 −1.78240
292292 1448.00i 0.290198i
293293 258.000i 0.0514421i 0.999669 + 0.0257210i 0.00818816π0.00818816\pi
−0.999669 + 0.0257210i 0.991812π0.991812\pi
294294 −5232.00 −1.03788
295295 0 0
296296 −272.000 −0.0534111
297297 − 960.000i − 0.187558i
298298 − 1500.00i − 0.291586i
299299 7656.00 1.48080
300300 0 0
301301 −128.000 −0.0245110
302302 − 896.000i − 0.170725i
303303 − 2064.00i − 0.391332i
304304 1600.00 0.301863
305305 0 0
306306 4884.00 0.912417
307307 − 8944.00i − 1.66274i −0.555720 0.831370i 0.687557π-0.687557\pi
0.555720 0.831370i 0.312443π-0.312443\pi
308308 192.000i 0.0355202i
309309 −7904.00 −1.45515
310310 0 0
311311 1392.00 0.253804 0.126902 0.991915i 0.459497π-0.459497\pi
0.126902 + 0.991915i 0.459497π0.459497\pi
312312 3712.00i 0.673560i
313313 5878.00i 1.06148i 0.847534 + 0.530742i 0.178087π0.178087\pi
−0.847534 + 0.530742i 0.821913π0.821913\pi
314314 −4492.00 −0.807319
315315 0 0
316316 −640.000 −0.113933
317317 10326.0i 1.82955i 0.403969 + 0.914773i 0.367630π0.367630\pi
−0.403969 + 0.914773i 0.632370π0.632370\pi
318318 3552.00i 0.626372i
319319 1080.00 0.189556
320320 0 0
321321 192.000 0.0333844
322322 − 1056.00i − 0.182760i
323323 6600.00i 1.13695i
324324 1436.00 0.246228
325325 0 0
326326 −1136.00 −0.192998
327327 − 7600.00i − 1.28526i
328328 3504.00i 0.589866i
329329 −816.000 −0.136740
330330 0 0
331331 −4228.00 −0.702090 −0.351045 0.936359i 0.614174π-0.614174\pi
−0.351045 + 0.936359i 0.614174π0.614174\pi
332332 288.000i 0.0476086i
333333 1258.00i 0.207021i
334334 3048.00 0.499339
335335 0 0
336336 512.000 0.0831306
337337 1106.00i 0.178776i 0.995997 + 0.0893882i 0.0284912π0.0284912\pi
−0.995997 + 0.0893882i 0.971509π0.971509\pi
338338 − 2334.00i − 0.375600i
339339 −8304.00 −1.33042
340340 0 0
341341 1824.00 0.289663
342342 − 7400.00i − 1.17002i
343343 − 2680.00i − 0.421885i
344344 −256.000 −0.0401238
345345 0 0
346346 7404.00 1.15041
347347 9336.00i 1.44433i 0.691720 + 0.722165i 0.256853π0.256853\pi
−0.691720 + 0.722165i 0.743147π0.743147\pi
348348 − 2880.00i − 0.443633i
349349 11770.0 1.80525 0.902627 0.430424i 0.141636π-0.141636\pi
0.902627 + 0.430424i 0.141636π0.141636\pi
350350 0 0
351351 4640.00 0.705598
352352 384.000i 0.0581456i
353353 − 8322.00i − 1.25477i −0.778707 0.627387i 0.784124π-0.784124\pi
0.778707 0.627387i 0.215876π-0.215876\pi
354354 6720.00 1.00894
355355 0 0
356356 3240.00 0.482359
357357 2112.00i 0.313106i
358358 − 6360.00i − 0.938929i
359359 −10680.0 −1.57011 −0.785054 0.619427i 0.787365π-0.787365\pi
−0.785054 + 0.619427i 0.787365π0.787365\pi
360360 0 0
361361 3141.00 0.457938
362362 − 4196.00i − 0.609218i
363363 − 9496.00i − 1.37303i
364364 −928.000 −0.133628
365365 0 0
366366 −14432.0 −2.06113
367367 − 5884.00i − 0.836900i −0.908240 0.418450i 0.862574π-0.862574\pi
0.908240 0.418450i 0.137426π-0.137426\pi
368368 − 2112.00i − 0.299173i
369369 16206.0 2.28632
370370 0 0
371371 −888.000 −0.124266
372372 − 4864.00i − 0.677921i
373373 2098.00i 0.291234i 0.989341 + 0.145617i 0.0465167π0.0465167\pi
−0.989341 + 0.145617i 0.953483π0.953483\pi
374374 −1584.00 −0.219002
375375 0 0
376376 −1632.00 −0.223840
377377 5220.00i 0.713113i
378378 − 640.000i − 0.0870848i
379379 −3860.00 −0.523153 −0.261576 0.965183i 0.584242π-0.584242\pi
−0.261576 + 0.965183i 0.584242π0.584242\pi
380380 0 0
381381 992.000 0.133390
382382 8784.00i 1.17651i
383383 9588.00i 1.27917i 0.768718 + 0.639587i 0.220895π0.220895\pi
−0.768718 + 0.639587i 0.779105π0.779105\pi
384384 1024.00 0.136083
385385 0 0
386386 −4316.00 −0.569116
387387 1184.00i 0.155520i
388388 − 4424.00i − 0.578852i
389389 13410.0 1.74785 0.873925 0.486060i 0.161566π-0.161566\pi
0.873925 + 0.486060i 0.161566π0.161566\pi
390390 0 0
391391 8712.00 1.12682
392392 − 2616.00i − 0.337061i
393393 1056.00i 0.135542i
394394 2148.00 0.274657
395395 0 0
396396 1776.00 0.225372
397397 − 13114.0i − 1.65787i −0.559348 0.828933i 0.688948π-0.688948\pi
0.559348 0.828933i 0.311052π-0.311052\pi
398398 − 5680.00i − 0.715358i
399399 3200.00 0.401505
400400 0 0
401401 −5838.00 −0.727022 −0.363511 0.931590i 0.618422π-0.618422\pi
−0.363511 + 0.931590i 0.618422π0.618422\pi
402402 16384.0i 2.03274i
403403 8816.00i 1.08972i
404404 1032.00 0.127089
405405 0 0
406406 720.000 0.0880123
407407 − 408.000i − 0.0496899i
408408 4224.00i 0.512547i
409409 −9530.00 −1.15215 −0.576074 0.817398i 0.695416π-0.695416\pi
−0.576074 + 0.817398i 0.695416π0.695416\pi
410410 0 0
411411 10032.0 1.20400
412412 − 3952.00i − 0.472575i
413413 1680.00i 0.200163i
414414 −9768.00 −1.15959
415415 0 0
416416 −1856.00 −0.218745
417417 22880.0i 2.68690i
418418 2400.00i 0.280832i
419419 −7260.00 −0.846478 −0.423239 0.906018i 0.639107π-0.639107\pi
−0.423239 + 0.906018i 0.639107π0.639107\pi
420420 0 0
421421 12062.0 1.39636 0.698178 0.715924i 0.253994π-0.253994\pi
0.698178 + 0.715924i 0.253994π0.253994\pi
422422 − 5336.00i − 0.615527i
423423 7548.00i 0.867604i
424424 −1776.00 −0.203420
425425 0 0
426426 −6912.00 −0.786121
427427 − 3608.00i − 0.408907i
428428 96.0000i 0.0108419i
429429 −5568.00 −0.626633
430430 0 0
431431 −13608.0 −1.52082 −0.760411 0.649442i 0.775002π-0.775002\pi
−0.760411 + 0.649442i 0.775002π0.775002\pi
432432 − 1280.00i − 0.142556i
433433 3838.00i 0.425964i 0.977056 + 0.212982i 0.0683176π0.0683176\pi
−0.977056 + 0.212982i 0.931682π0.931682\pi
434434 1216.00 0.134493
435435 0 0
436436 3800.00 0.417401
437437 − 13200.0i − 1.44495i
438438 5792.00i 0.631855i
439439 −7400.00 −0.804516 −0.402258 0.915526i 0.631775π-0.631775\pi
−0.402258 + 0.915526i 0.631775π0.631775\pi
440440 0 0
441441 −12099.0 −1.30645
442442 − 7656.00i − 0.823889i
443443 − 8352.00i − 0.895746i −0.894097 0.447873i 0.852182π-0.852182\pi
0.894097 0.447873i 0.147818π-0.147818\pi
444444 −1088.00 −0.116293
445445 0 0
446446 3544.00 0.376263
447447 − 6000.00i − 0.634878i
448448 256.000i 0.0269975i
449449 −10770.0 −1.13200 −0.566000 0.824405i 0.691510π-0.691510\pi
−0.566000 + 0.824405i 0.691510π0.691510\pi
450450 0 0
451451 −5256.00 −0.548770
452452 − 4152.00i − 0.432066i
453453 − 3584.00i − 0.371724i
454454 5568.00 0.575593
455455 0 0
456456 6400.00 0.657253
457457 − 6694.00i − 0.685191i −0.939483 0.342595i 0.888694π-0.888694\pi
0.939483 0.342595i 0.111306π-0.111306\pi
458458 − 700.000i − 0.0714167i
459459 5280.00 0.536927
460460 0 0
461461 −3018.00 −0.304907 −0.152454 0.988311i 0.548717π-0.548717\pi
−0.152454 + 0.988311i 0.548717π0.548717\pi
462462 768.000i 0.0773389i
463463 − 14492.0i − 1.45464i −0.686296 0.727322i 0.740765π-0.740765\pi
0.686296 0.727322i 0.259235π-0.259235\pi
464464 1440.00 0.144074
465465 0 0
466466 3924.00 0.390077
467467 7776.00i 0.770515i 0.922809 + 0.385257i 0.125887π0.125887\pi
−0.922809 + 0.385257i 0.874113π0.874113\pi
468468 8584.00i 0.847854i
469469 −4096.00 −0.403274
470470 0 0
471471 −17968.0 −1.75780
472472 3360.00i 0.327662i
473473 − 384.000i − 0.0373284i
474474 −2560.00 −0.248069
475475 0 0
476476 −1056.00 −0.101684
477477 8214.00i 0.788455i
478478 8640.00i 0.826746i
479479 13680.0 1.30492 0.652458 0.757825i 0.273738π-0.273738\pi
0.652458 + 0.757825i 0.273738π0.273738\pi
480480 0 0
481481 1972.00 0.186934
482482 − 956.000i − 0.0903415i
483483 − 4224.00i − 0.397927i
484484 4748.00 0.445905
485485 0 0
486486 10064.0 0.939326
487487 7916.00i 0.736567i 0.929714 + 0.368284i 0.120054π0.120054\pi
−0.929714 + 0.368284i 0.879946π0.879946\pi
488488 − 7216.00i − 0.669371i
489489 −4544.00 −0.420218
490490 0 0
491491 13932.0 1.28053 0.640267 0.768152i 0.278824π-0.278824\pi
0.640267 + 0.768152i 0.278824π0.278824\pi
492492 14016.0i 1.28433i
493493 5940.00i 0.542645i
494494 −11600.0 −1.05650
495495 0 0
496496 2432.00 0.220161
497497 − 1728.00i − 0.155959i
498498 1152.00i 0.103659i
499499 8260.00 0.741019 0.370509 0.928829i 0.379183π-0.379183\pi
0.370509 + 0.928829i 0.379183π0.379183\pi
500500 0 0
501501 12192.0 1.08722
502502 5304.00i 0.471572i
503503 11148.0i 0.988200i 0.869405 + 0.494100i 0.164502π0.164502\pi
−0.869405 + 0.494100i 0.835498π0.835498\pi
504504 1184.00 0.104642
505505 0 0
506506 3168.00 0.278330
507507 − 9336.00i − 0.817803i
508508 496.000i 0.0433198i
509509 9690.00 0.843815 0.421907 0.906639i 0.361361π-0.361361\pi
0.421907 + 0.906639i 0.361361π0.361361\pi
510510 0 0
511511 −1448.00 −0.125354
512512 512.000i 0.0441942i
513513 − 8000.00i − 0.688516i
514514 4668.00 0.400577
515515 0 0
516516 −1024.00 −0.0873626
517517 − 2448.00i − 0.208245i
518518 − 272.000i − 0.0230714i
519519 29616.0 2.50481
520520 0 0
521521 −16038.0 −1.34863 −0.674316 0.738443i 0.735562π-0.735562\pi
−0.674316 + 0.738443i 0.735562π0.735562\pi
522522 − 6660.00i − 0.558430i
523523 − 992.000i − 0.0829391i −0.999140 0.0414695i 0.986796π-0.986796\pi
0.999140 0.0414695i 0.0132039π-0.0132039\pi
524524 −528.000 −0.0440187
525525 0 0
526526 −7896.00 −0.654528
527527 10032.0i 0.829223i
528528 1536.00i 0.126602i
529529 −5257.00 −0.432070
530530 0 0
531531 15540.0 1.27002
532532 1600.00i 0.130392i
533533 − 25404.0i − 2.06448i
534534 12960.0 1.05025
535535 0 0
536536 −8192.00 −0.660150
537537 − 25440.0i − 2.04435i
538538 − 3180.00i − 0.254832i
539539 3924.00 0.313578
540540 0 0
541541 7142.00 0.567576 0.283788 0.958887i 0.408409π-0.408409\pi
0.283788 + 0.958887i 0.408409π0.408409\pi
542542 9904.00i 0.784895i
543543 − 16784.0i − 1.32646i
544544 −2112.00 −0.166455
545545 0 0
546546 −3712.00 −0.290950
547547 7616.00i 0.595314i 0.954673 + 0.297657i 0.0962051π0.0962051\pi
−0.954673 + 0.297657i 0.903795π0.903795\pi
548548 5016.00i 0.391009i
549549 −33374.0 −2.59448
550550 0 0
551551 9000.00 0.695849
552552 − 8448.00i − 0.651396i
553553 − 640.000i − 0.0492144i
554554 −3292.00 −0.252462
555555 0 0
556556 −11440.0 −0.872597
557557 − 10314.0i − 0.784593i −0.919839 0.392296i 0.871681π-0.871681\pi
0.919839 0.392296i 0.128319π-0.128319\pi
558558 − 11248.0i − 0.853344i
559559 1856.00 0.140430
560560 0 0
561561 −6336.00 −0.476838
562562 − 2316.00i − 0.173834i
563563 7128.00i 0.533587i 0.963754 + 0.266793i 0.0859641π0.0859641\pi
−0.963754 + 0.266793i 0.914036π0.914036\pi
564564 −6528.00 −0.487373
565565 0 0
566566 13984.0 1.03850
567567 1436.00i 0.106360i
568568 − 3456.00i − 0.255300i
569569 −2010.00 −0.148091 −0.0740453 0.997255i 0.523591π-0.523591\pi
−0.0740453 + 0.997255i 0.523591π0.523591\pi
570570 0 0
571571 −23188.0 −1.69945 −0.849726 0.527224i 0.823233π-0.823233\pi
−0.849726 + 0.527224i 0.823233π0.823233\pi
572572 − 2784.00i − 0.203505i
573573 35136.0i 2.56165i
574574 −3504.00 −0.254798
575575 0 0
576576 2368.00 0.171296
577577 22466.0i 1.62092i 0.585793 + 0.810461i 0.300783π0.300783\pi
−0.585793 + 0.810461i 0.699217π0.699217\pi
578578 1114.00i 0.0801666i
579579 −17264.0 −1.23915
580580 0 0
581581 −288.000 −0.0205650
582582 − 17696.0i − 1.26035i
583583 − 2664.00i − 0.189248i
584584 −2896.00 −0.205201
585585 0 0
586586 −516.000 −0.0363750
587587 22776.0i 1.60148i 0.599015 + 0.800738i 0.295559π0.295559\pi
−0.599015 + 0.800738i 0.704441π0.704441\pi
588588 − 10464.0i − 0.733891i
589589 15200.0 1.06334
590590 0 0
591591 8592.00 0.598016
592592 − 544.000i − 0.0377673i
593593 21198.0i 1.46796i 0.679174 + 0.733978i 0.262338π0.262338\pi
−0.679174 + 0.733978i 0.737662π0.737662\pi
594594 1920.00 0.132624
595595 0 0
596596 3000.00 0.206183
597597 − 22720.0i − 1.55757i
598598 15312.0i 1.04708i
599599 −15960.0 −1.08866 −0.544330 0.838871i 0.683216π-0.683216\pi
−0.544330 + 0.838871i 0.683216π0.683216\pi
600600 0 0
601601 5882.00 0.399221 0.199610 0.979875i 0.436032π-0.436032\pi
0.199610 + 0.979875i 0.436032π0.436032\pi
602602 − 256.000i − 0.0173319i
603603 37888.0i 2.55874i
604604 1792.00 0.120721
605605 0 0
606606 4128.00 0.276714
607607 8516.00i 0.569446i 0.958610 + 0.284723i 0.0919016π0.0919016\pi
−0.958610 + 0.284723i 0.908098π0.908098\pi
608608 3200.00i 0.213449i
609609 2880.00 0.191631
610610 0 0
611611 11832.0 0.783423
612612 9768.00i 0.645176i
613613 − 8462.00i − 0.557548i −0.960357 0.278774i 0.910072π-0.910072\pi
0.960357 0.278774i 0.0899280π-0.0899280\pi
614614 17888.0 1.17573
615615 0 0
616616 −384.000 −0.0251166
617617 − 11094.0i − 0.723870i −0.932203 0.361935i 0.882116π-0.882116\pi
0.932203 0.361935i 0.117884π-0.117884\pi
618618 − 15808.0i − 1.02895i
619619 −2180.00 −0.141553 −0.0707767 0.997492i 0.522548π-0.522548\pi
−0.0707767 + 0.997492i 0.522548π0.522548\pi
620620 0 0
621621 −10560.0 −0.682380
622622 2784.00i 0.179467i
623623 3240.00i 0.208359i
624624 −7424.00 −0.476279
625625 0 0
626626 −11756.0 −0.750582
627627 9600.00i 0.611463i
628628 − 8984.00i − 0.570861i
629629 2244.00 0.142248
630630 0 0
631631 −26848.0 −1.69382 −0.846911 0.531734i 0.821541π-0.821541\pi
−0.846911 + 0.531734i 0.821541π0.821541\pi
632632 − 1280.00i − 0.0805628i
633633 − 21344.0i − 1.34020i
634634 −20652.0 −1.29368
635635 0 0
636636 −7104.00 −0.442912
637637 18966.0i 1.17969i
638638 2160.00i 0.134036i
639639 −15984.0 −0.989542
640640 0 0
641641 26322.0 1.62193 0.810965 0.585095i 0.198943π-0.198943\pi
0.810965 + 0.585095i 0.198943π0.198943\pi
642642 384.000i 0.0236063i
643643 10168.0i 0.623619i 0.950145 + 0.311809i 0.100935π0.100935\pi
−0.950145 + 0.311809i 0.899065π0.899065\pi
644644 2112.00 0.129231
645645 0 0
646646 −13200.0 −0.803943
647647 − 23604.0i − 1.43426i −0.696937 0.717132i 0.745454π-0.745454\pi
0.696937 0.717132i 0.254546π-0.254546\pi
648648 2872.00i 0.174109i
649649 −5040.00 −0.304834
650650 0 0
651651 4864.00 0.292834
652652 − 2272.00i − 0.136470i
653653 − 16422.0i − 0.984139i −0.870556 0.492069i 0.836241π-0.836241\pi
0.870556 0.492069i 0.163759π-0.163759\pi
654654 15200.0 0.908818
655655 0 0
656656 −7008.00 −0.417098
657657 13394.0i 0.795357i
658658 − 1632.00i − 0.0966899i
659659 26100.0 1.54281 0.771405 0.636345i 0.219554π-0.219554\pi
0.771405 + 0.636345i 0.219554π0.219554\pi
660660 0 0
661661 −3058.00 −0.179943 −0.0899716 0.995944i 0.528678π-0.528678\pi
−0.0899716 + 0.995944i 0.528678π0.528678\pi
662662 − 8456.00i − 0.496453i
663663 − 30624.0i − 1.79387i
664664 −576.000 −0.0336644
665665 0 0
666666 −2516.00 −0.146386
667667 − 11880.0i − 0.689648i
668668 6096.00i 0.353086i
669669 14176.0 0.819246
670670 0 0
671671 10824.0 0.622736
672672 1024.00i 0.0587822i
673673 − 10802.0i − 0.618702i −0.950948 0.309351i 0.899888π-0.899888\pi
0.950948 0.309351i 0.100112π-0.100112\pi
674674 −2212.00 −0.126414
675675 0 0
676676 4668.00 0.265589
677677 − 10674.0i − 0.605960i −0.952997 0.302980i 0.902018π-0.902018\pi
0.952997 0.302980i 0.0979816π-0.0979816\pi
678678 − 16608.0i − 0.940747i
679679 4424.00 0.250041
680680 0 0
681681 22272.0 1.25325
682682 3648.00i 0.204823i
683683 28608.0i 1.60272i 0.598185 + 0.801358i 0.295889π0.295889\pi
−0.598185 + 0.801358i 0.704111π0.704111\pi
684684 14800.0 0.827328
685685 0 0
686686 5360.00 0.298317
687687 − 2800.00i − 0.155497i
688688 − 512.000i − 0.0283718i
689689 12876.0 0.711954
690690 0 0
691691 −2428.00 −0.133669 −0.0668346 0.997764i 0.521290π-0.521290\pi
−0.0668346 + 0.997764i 0.521290π0.521290\pi
692692 14808.0i 0.813462i
693693 1776.00i 0.0973516i
694694 −18672.0 −1.02130
695695 0 0
696696 5760.00 0.313696
697697 − 28908.0i − 1.57097i
698698 23540.0i 1.27651i
699699 15696.0 0.849324
700700 0 0
701701 −6618.00 −0.356574 −0.178287 0.983979i 0.557056π-0.557056\pi
−0.178287 + 0.983979i 0.557056π0.557056\pi
702702 9280.00i 0.498933i
703703 − 3400.00i − 0.182409i
704704 −768.000 −0.0411152
705705 0 0
706706 16644.0 0.887259
707707 1032.00i 0.0548972i
708708 13440.0i 0.713427i
709709 −20510.0 −1.08642 −0.543208 0.839598i 0.682791π-0.682791\pi
−0.543208 + 0.839598i 0.682791π0.682791\pi
710710 0 0
711711 −5920.00 −0.312261
712712 6480.00i 0.341079i
713713 − 20064.0i − 1.05386i
714714 −4224.00 −0.221399
715715 0 0
716716 12720.0 0.663923
717717 34560.0i 1.80009i
718718 − 21360.0i − 1.11023i
719719 −31680.0 −1.64321 −0.821603 0.570061i 0.806920π-0.806920\pi
−0.821603 + 0.570061i 0.806920π0.806920\pi
720720 0 0
721721 3952.00 0.204133
722722 6282.00i 0.323811i
723723 − 3824.00i − 0.196703i
724724 8392.00 0.430782
725725 0 0
726726 18992.0 0.970880
727727 13196.0i 0.673195i 0.941649 + 0.336597i 0.109276π0.109276\pi
−0.941649 + 0.336597i 0.890724π0.890724\pi
728728 − 1856.00i − 0.0944889i
729729 30563.0 1.55276
730730 0 0
731731 2112.00 0.106861
732732 − 28864.0i − 1.45744i
733733 − 8102.00i − 0.408259i −0.978944 0.204130i 0.934564π-0.934564\pi
0.978944 0.204130i 0.0654364π-0.0654364\pi
734734 11768.0 0.591778
735735 0 0
736736 4224.00 0.211547
737737 − 12288.0i − 0.614158i
738738 32412.0i 1.61667i
739739 12580.0 0.626201 0.313101 0.949720i 0.398632π-0.398632\pi
0.313101 + 0.949720i 0.398632π0.398632\pi
740740 0 0
741741 −46400.0 −2.30033
742742 − 1776.00i − 0.0878693i
743743 − 29892.0i − 1.47595i −0.674828 0.737975i 0.735782π-0.735782\pi
0.674828 0.737975i 0.264218π-0.264218\pi
744744 9728.00 0.479363
745745 0 0
746746 −4196.00 −0.205934
747747 2664.00i 0.130483i
748748 − 3168.00i − 0.154858i
749749 −96.0000 −0.00468326
750750 0 0
751751 −40408.0 −1.96339 −0.981697 0.190450i 0.939005π-0.939005\pi
−0.981697 + 0.190450i 0.939005π0.939005\pi
752752 − 3264.00i − 0.158279i
753753 21216.0i 1.02676i
754754 −10440.0 −0.504247
755755 0 0
756756 1280.00 0.0615782
757757 32366.0i 1.55398i 0.629513 + 0.776990i 0.283254π0.283254\pi
−0.629513 + 0.776990i 0.716746π0.716746\pi
758758 − 7720.00i − 0.369925i
759759 12672.0 0.606014
760760 0 0
761761 −17238.0 −0.821126 −0.410563 0.911832i 0.634668π-0.634668\pi
−0.410563 + 0.911832i 0.634668π0.634668\pi
762762 1984.00i 0.0943212i
763763 3800.00i 0.180300i
764764 −17568.0 −0.831921
765765 0 0
766766 −19176.0 −0.904513
767767 − 24360.0i − 1.14679i
768768 2048.00i 0.0962250i
769769 −10850.0 −0.508792 −0.254396 0.967100i 0.581877π-0.581877\pi
−0.254396 + 0.967100i 0.581877π0.581877\pi
770770 0 0
771771 18672.0 0.872186
772772 − 8632.00i − 0.402425i
773773 − 9102.00i − 0.423514i −0.977322 0.211757i 0.932081π-0.932081\pi
0.977322 0.211757i 0.0679185π-0.0679185\pi
774774 −2368.00 −0.109969
775775 0 0
776776 8848.00 0.409310
777777 − 1088.00i − 0.0502340i
778778 26820.0i 1.23592i
779779 −43800.0 −2.01450
780780 0 0
781781 5184.00 0.237514
782782 17424.0i 0.796779i
783783 − 7200.00i − 0.328617i
784784 5232.00 0.238338
785785 0 0
786786 −2112.00 −0.0958429
787787 − 25504.0i − 1.15517i −0.816330 0.577585i 0.803995π-0.803995\pi
0.816330 0.577585i 0.196005π-0.196005\pi
788788 4296.00i 0.194212i
789789 −31584.0 −1.42512
790790 0 0
791791 4152.00 0.186635
792792 3552.00i 0.159362i
793793 52316.0i 2.34274i
794794 26228.0 1.17229
795795 0 0
796796 11360.0 0.505835
797797 14166.0i 0.629593i 0.949159 + 0.314796i 0.101936π0.101936\pi
−0.949159 + 0.314796i 0.898064π0.898064\pi
798798 6400.00i 0.283907i
799799 13464.0 0.596148
800800 0 0
801801 29970.0 1.32202
802802 − 11676.0i − 0.514082i
803803 − 4344.00i − 0.190905i
804804 −32768.0 −1.43736
805805 0 0
806806 −17632.0 −0.770547
807807 − 12720.0i − 0.554852i
808808 2064.00i 0.0898654i
809809 −33210.0 −1.44327 −0.721633 0.692276i 0.756608π-0.756608\pi
−0.721633 + 0.692276i 0.756608π0.756608\pi
810810 0 0
811811 39212.0 1.69780 0.848902 0.528550i 0.177264π-0.177264\pi
0.848902 + 0.528550i 0.177264π0.177264\pi
812812 1440.00i 0.0622341i
813813 39616.0i 1.70897i
814814 816.000 0.0351361
815815 0 0
816816 −8448.00 −0.362425
817817 − 3200.00i − 0.137030i
818818 − 19060.0i − 0.814691i
819819 −8584.00 −0.366238
820820 0 0
821821 6222.00 0.264494 0.132247 0.991217i 0.457781π-0.457781\pi
0.132247 + 0.991217i 0.457781π0.457781\pi
822822 20064.0i 0.851353i
823823 − 31172.0i − 1.32028i −0.751144 0.660138i 0.770498π-0.770498\pi
0.751144 0.660138i 0.229502π-0.229502\pi
824824 7904.00 0.334161
825825 0 0
826826 −3360.00 −0.141537
827827 − 264.000i − 0.0111006i −0.999985 0.00555029i 0.998233π-0.998233\pi
0.999985 0.00555029i 0.00176672π-0.00176672\pi
828828 − 19536.0i − 0.819955i
829829 29050.0 1.21707 0.608533 0.793528i 0.291758π-0.291758\pi
0.608533 + 0.793528i 0.291758π0.291758\pi
830830 0 0
831831 −13168.0 −0.549691
832832 − 3712.00i − 0.154676i
833833 21582.0i 0.897685i
834834 −45760.0 −1.89993
835835 0 0
836836 −4800.00 −0.198578
837837 − 12160.0i − 0.502164i
838838 − 14520.0i − 0.598550i
839839 21720.0 0.893752 0.446876 0.894596i 0.352537π-0.352537\pi
0.446876 + 0.894596i 0.352537π0.352537\pi
840840 0 0
841841 −16289.0 −0.667883
842842 24124.0i 0.987373i
843843 − 9264.00i − 0.378492i
844844 10672.0 0.435243
845845 0 0
846846 −15096.0 −0.613488
847847 4748.00i 0.192613i
848848 − 3552.00i − 0.143840i
849849 55936.0 2.26115
850850 0 0
851851 −4488.00 −0.180783
852852 − 13824.0i − 0.555871i
853853 6658.00i 0.267252i 0.991032 + 0.133626i 0.0426620π0.0426620\pi
−0.991032 + 0.133626i 0.957338π0.957338\pi
854854 7216.00 0.289141
855855 0 0
856856 −192.000 −0.00766638
857857 − 13974.0i − 0.556993i −0.960437 0.278496i 0.910164π-0.910164\pi
0.960437 0.278496i 0.0898360π-0.0898360\pi
858858 − 11136.0i − 0.443096i
859859 −23780.0 −0.944544 −0.472272 0.881453i 0.656566π-0.656566\pi
−0.472272 + 0.881453i 0.656566π0.656566\pi
860860 0 0
861861 −14016.0 −0.554778
862862 − 27216.0i − 1.07538i
863863 12228.0i 0.482324i 0.970485 + 0.241162i 0.0775286π0.0775286\pi
−0.970485 + 0.241162i 0.922471π0.922471\pi
864864 2560.00 0.100802
865865 0 0
866866 −7676.00 −0.301202
867867 4456.00i 0.174549i
868868 2432.00i 0.0951008i
869869 1920.00 0.0749500
870870 0 0
871871 59392.0 2.31047
872872 7600.00i 0.295147i
873873 − 40922.0i − 1.58648i
874874 26400.0 1.02173
875875 0 0
876876 −11584.0 −0.446789
877877 11606.0i 0.446872i 0.974719 + 0.223436i 0.0717274π0.0717274\pi
−0.974719 + 0.223436i 0.928273π0.928273\pi
878878 − 14800.0i − 0.568879i
879879 −2064.00 −0.0792002
880880 0 0
881881 −32958.0 −1.26037 −0.630183 0.776446i 0.717020π-0.717020\pi
−0.630183 + 0.776446i 0.717020π0.717020\pi
882882 − 24198.0i − 0.923797i
883883 − 8072.00i − 0.307638i −0.988099 0.153819i 0.950843π-0.950843\pi
0.988099 0.153819i 0.0491573π-0.0491573\pi
884884 15312.0 0.582577
885885 0 0
886886 16704.0 0.633388
887887 15756.0i 0.596431i 0.954498 + 0.298216i 0.0963915π0.0963915\pi
−0.954498 + 0.298216i 0.903609π0.903609\pi
888888 − 2176.00i − 0.0822317i
889889 −496.000 −0.0187124
890890 0 0
891891 −4308.00 −0.161979
892892 7088.00i 0.266058i
893893 − 20400.0i − 0.764457i
894894 12000.0 0.448926
895895 0 0
896896 −512.000 −0.0190901
897897 61248.0i 2.27983i
898898 − 21540.0i − 0.800444i
899899 13680.0 0.507512
900900 0 0
901901 14652.0 0.541763
902902 − 10512.0i − 0.388039i
903903 − 1024.00i − 0.0377371i
904904 8304.00 0.305517
905905 0 0
906906 7168.00 0.262849
907907 18776.0i 0.687372i 0.939085 + 0.343686i 0.111676π0.111676\pi
−0.939085 + 0.343686i 0.888324π0.888324\pi
908908 11136.0i 0.407006i
909909 9546.00 0.348318
910910 0 0
911911 −20568.0 −0.748022 −0.374011 0.927424i 0.622018π-0.622018\pi
−0.374011 + 0.927424i 0.622018π0.622018\pi
912912 12800.0i 0.464748i
913913 − 864.000i − 0.0313190i
914914 13388.0 0.484503
915915 0 0
916916 1400.00 0.0504992
917917 − 528.000i − 0.0190143i
918918 10560.0i 0.379664i
919919 6280.00 0.225417 0.112708 0.993628i 0.464047π-0.464047\pi
0.112708 + 0.993628i 0.464047π0.464047\pi
920920 0 0
921921 71552.0 2.55996
922922 − 6036.00i − 0.215602i
923923 25056.0i 0.893530i
924924 −1536.00 −0.0546869
925925 0 0
926926 28984.0 1.02859
927927 − 36556.0i − 1.29521i
928928 2880.00i 0.101876i
929929 20430.0 0.721514 0.360757 0.932660i 0.382518π-0.382518\pi
0.360757 + 0.932660i 0.382518π0.382518\pi
930930 0 0
931931 32700.0 1.15113
932932 7848.00i 0.275826i
933933 11136.0i 0.390757i
934934 −15552.0 −0.544836
935935 0 0
936936 −17168.0 −0.599523
937937 8906.00i 0.310508i 0.987875 + 0.155254i 0.0496197π0.0496197\pi
−0.987875 + 0.155254i 0.950380π0.950380\pi
938938 − 8192.00i − 0.285158i
939939 −47024.0 −1.63426
940940 0 0
941941 −17418.0 −0.603412 −0.301706 0.953401i 0.597556π-0.597556\pi
−0.301706 + 0.953401i 0.597556π0.597556\pi
942942 − 35936.0i − 1.24295i
943943 57816.0i 1.99655i
944944 −6720.00 −0.231692
945945 0 0
946946 768.000 0.0263952
947947 − 2544.00i − 0.0872956i −0.999047 0.0436478i 0.986102π-0.986102\pi
0.999047 0.0436478i 0.0138979π-0.0138979\pi
948948 − 5120.00i − 0.175411i
949949 20996.0 0.718187
950950 0 0
951951 −82608.0 −2.81677
952952 − 2112.00i − 0.0719016i
953953 − 15402.0i − 0.523525i −0.965132 0.261763i 0.915696π-0.915696\pi
0.965132 0.261763i 0.0843038π-0.0843038\pi
954954 −16428.0 −0.557522
955955 0 0
956956 −17280.0 −0.584597
957957 8640.00i 0.291841i
958958 27360.0i 0.922716i
959959 −5016.00 −0.168900
960960 0 0
961961 −6687.00 −0.224464
962962 3944.00i 0.132183i
963963 888.000i 0.0297148i
964964 1912.00 0.0638811
965965 0 0
966966 8448.00 0.281377
967967 − 49444.0i − 1.64427i −0.569291 0.822136i 0.692782π-0.692782\pi
0.569291 0.822136i 0.307218π-0.307218\pi
968968 9496.00i 0.315303i
969969 −52800.0 −1.75044
970970 0 0
971971 −25188.0 −0.832463 −0.416231 0.909259i 0.636649π-0.636649\pi
−0.416231 + 0.909259i 0.636649π0.636649\pi
972972 20128.0i 0.664204i
973973 − 11440.0i − 0.376927i
974974 −15832.0 −0.520832
975975 0 0
976976 14432.0 0.473317
977977 2946.00i 0.0964697i 0.998836 + 0.0482348i 0.0153596π0.0153596\pi
−0.998836 + 0.0482348i 0.984640π0.984640\pi
978978 − 9088.00i − 0.297139i
979979 −9720.00 −0.317316
980980 0 0
981981 35150.0 1.14399
982982 27864.0i 0.905475i
983983 − 15012.0i − 0.487089i −0.969890 0.243544i 0.921690π-0.921690\pi
0.969890 0.243544i 0.0783102π-0.0783102\pi
984984 −28032.0 −0.908158
985985 0 0
986986 −11880.0 −0.383708
987987 − 6528.00i − 0.210525i
988988 − 23200.0i − 0.747055i
989989 −4224.00 −0.135809
990990 0 0
991991 −5128.00 −0.164376 −0.0821878 0.996617i 0.526191π-0.526191\pi
−0.0821878 + 0.996617i 0.526191π0.526191\pi
992992 4864.00i 0.155678i
993993 − 33824.0i − 1.08094i
994994 3456.00 0.110279
995995 0 0
996996 −2304.00 −0.0732982
997997 − 49714.0i − 1.57920i −0.613625 0.789598i 0.710289π-0.710289\pi
0.613625 0.789598i 0.289711π-0.289711\pi
998998 16520.0i 0.523979i
999999 −2720.00 −0.0861431
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 50.4.b.a.49.2 2
3.2 odd 2 450.4.c.d.199.1 2
4.3 odd 2 400.4.c.c.49.1 2
5.2 odd 4 50.4.a.c.1.1 1
5.3 odd 4 10.4.a.a.1.1 1
5.4 even 2 inner 50.4.b.a.49.1 2
15.2 even 4 450.4.a.q.1.1 1
15.8 even 4 90.4.a.a.1.1 1
15.14 odd 2 450.4.c.d.199.2 2
20.3 even 4 80.4.a.f.1.1 1
20.7 even 4 400.4.a.b.1.1 1
20.19 odd 2 400.4.c.c.49.2 2
35.3 even 12 490.4.e.a.471.1 2
35.13 even 4 490.4.a.o.1.1 1
35.18 odd 12 490.4.e.i.471.1 2
35.23 odd 12 490.4.e.i.361.1 2
35.27 even 4 2450.4.a.b.1.1 1
35.33 even 12 490.4.e.a.361.1 2
40.3 even 4 320.4.a.b.1.1 1
40.13 odd 4 320.4.a.m.1.1 1
40.27 even 4 1600.4.a.bx.1.1 1
40.37 odd 4 1600.4.a.d.1.1 1
45.13 odd 12 810.4.e.c.541.1 2
45.23 even 12 810.4.e.w.541.1 2
45.38 even 12 810.4.e.w.271.1 2
45.43 odd 12 810.4.e.c.271.1 2
55.43 even 4 1210.4.a.b.1.1 1
60.23 odd 4 720.4.a.j.1.1 1
65.38 odd 4 1690.4.a.a.1.1 1
80.3 even 4 1280.4.d.g.641.2 2
80.13 odd 4 1280.4.d.j.641.1 2
80.43 even 4 1280.4.d.g.641.1 2
80.53 odd 4 1280.4.d.j.641.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.a.a.1.1 1 5.3 odd 4
50.4.a.c.1.1 1 5.2 odd 4
50.4.b.a.49.1 2 5.4 even 2 inner
50.4.b.a.49.2 2 1.1 even 1 trivial
80.4.a.f.1.1 1 20.3 even 4
90.4.a.a.1.1 1 15.8 even 4
320.4.a.b.1.1 1 40.3 even 4
320.4.a.m.1.1 1 40.13 odd 4
400.4.a.b.1.1 1 20.7 even 4
400.4.c.c.49.1 2 4.3 odd 2
400.4.c.c.49.2 2 20.19 odd 2
450.4.a.q.1.1 1 15.2 even 4
450.4.c.d.199.1 2 3.2 odd 2
450.4.c.d.199.2 2 15.14 odd 2
490.4.a.o.1.1 1 35.13 even 4
490.4.e.a.361.1 2 35.33 even 12
490.4.e.a.471.1 2 35.3 even 12
490.4.e.i.361.1 2 35.23 odd 12
490.4.e.i.471.1 2 35.18 odd 12
720.4.a.j.1.1 1 60.23 odd 4
810.4.e.c.271.1 2 45.43 odd 12
810.4.e.c.541.1 2 45.13 odd 12
810.4.e.w.271.1 2 45.38 even 12
810.4.e.w.541.1 2 45.23 even 12
1210.4.a.b.1.1 1 55.43 even 4
1280.4.d.g.641.1 2 80.43 even 4
1280.4.d.g.641.2 2 80.3 even 4
1280.4.d.j.641.1 2 80.13 odd 4
1280.4.d.j.641.2 2 80.53 odd 4
1600.4.a.d.1.1 1 40.37 odd 4
1600.4.a.bx.1.1 1 40.27 even 4
1690.4.a.a.1.1 1 65.38 odd 4
2450.4.a.b.1.1 1 35.27 even 4