Properties

Label 650.6.b.k.599.8
Level 650650
Weight 66
Character 650.599
Analytic conductor 104.249104.249
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 650=25213 650 = 2 \cdot 5^{2} \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 650.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: 104.249482878104.249482878
Analytic rank: 00
Dimension: 88
Coefficient field: Q[x]/(x8+)\mathbb{Q}[x]/(x^{8} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8+1094x6+343849x4+27023076x2+498182400 x^{8} + 1094x^{6} + 343849x^{4} + 27023076x^{2} + 498182400 Copy content Toggle raw display
Coefficient ring: Z[a1,,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 23 2^{3}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 599.8
Root 20.3395i20.3395i of defining polynomial
Character χ\chi == 650.599
Dual form 650.6.b.k.599.1

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+4.00000iq2+25.3395iq316.0000q4101.358q6154.564iq764.0000iq8399.089q963.7546q11405.431iq12169.000iq13+618.256q14+256.000q16374.136iq171596.35iq18+42.3586q19+3916.57q21255.018iq221131.66iq23+1621.73q24+676.000q263955.20iq27+2473.03iq282487.62q29+9742.11q31+1024.00iq321615.51iq33+1496.54q34+6385.42q365341.14iq37+169.435iq38+4282.37q397131.82q41+15666.3iq42+6754.09iq43+1020.07q44+4526.62q46+14373.8iq47+6486.90iq487083.06q49+9480.40q51+2704.00iq52+13363.6iq53+15820.8q549892.10q56+1073.35iq579950.47iq5838502.8q59+28521.0q61+38968.4iq62+61684.8iq634096.00q64+6462.03q66+43953.6iq67+5986.17iq68+28675.6q69+40991.3q71+25541.7iq72+68431.1iq73+21364.6q74677.738q76+9854.17iq77+17129.5iq7820233.0q79+3244.21q8128527.3iq82+63147.4iq8362665.2q8427016.4q8663034.9iq87+4080.29iq88+116225.q8926121.3q91+18106.5iq92+246860.iq9357495.1q9425947.6q9659187.1iq9728332.2iq98+25443.7q99+O(q100)q+4.00000i q^{2} +25.3395i q^{3} -16.0000 q^{4} -101.358 q^{6} -154.564i q^{7} -64.0000i q^{8} -399.089 q^{9} -63.7546 q^{11} -405.431i q^{12} -169.000i q^{13} +618.256 q^{14} +256.000 q^{16} -374.136i q^{17} -1596.35i q^{18} +42.3586 q^{19} +3916.57 q^{21} -255.018i q^{22} -1131.66i q^{23} +1621.73 q^{24} +676.000 q^{26} -3955.20i q^{27} +2473.03i q^{28} -2487.62 q^{29} +9742.11 q^{31} +1024.00i q^{32} -1615.51i q^{33} +1496.54 q^{34} +6385.42 q^{36} -5341.14i q^{37} +169.435i q^{38} +4282.37 q^{39} -7131.82 q^{41} +15666.3i q^{42} +6754.09i q^{43} +1020.07 q^{44} +4526.62 q^{46} +14373.8i q^{47} +6486.90i q^{48} -7083.06 q^{49} +9480.40 q^{51} +2704.00i q^{52} +13363.6i q^{53} +15820.8 q^{54} -9892.10 q^{56} +1073.35i q^{57} -9950.47i q^{58} -38502.8 q^{59} +28521.0 q^{61} +38968.4i q^{62} +61684.8i q^{63} -4096.00 q^{64} +6462.03 q^{66} +43953.6i q^{67} +5986.17i q^{68} +28675.6 q^{69} +40991.3 q^{71} +25541.7i q^{72} +68431.1i q^{73} +21364.6 q^{74} -677.738 q^{76} +9854.17i q^{77} +17129.5i q^{78} -20233.0 q^{79} +3244.21 q^{81} -28527.3i q^{82} +63147.4i q^{83} -62665.2 q^{84} -27016.4 q^{86} -63034.9i q^{87} +4080.29i q^{88} +116225. q^{89} -26121.3 q^{91} +18106.5i q^{92} +246860. i q^{93} -57495.1 q^{94} -25947.6 q^{96} -59187.1i q^{97} -28332.2i q^{98} +25443.7 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q128q4160q6444q9588q11+1424q14+2048q16928q192016q21+2560q24+5408q26+9088q29+11236q31+14448q34+7104q36+6760q39+89060q99+O(q100) 8 q - 128 q^{4} - 160 q^{6} - 444 q^{9} - 588 q^{11} + 1424 q^{14} + 2048 q^{16} - 928 q^{19} - 2016 q^{21} + 2560 q^{24} + 5408 q^{26} + 9088 q^{29} + 11236 q^{31} + 14448 q^{34} + 7104 q^{36} + 6760 q^{39}+ \cdots - 89060 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/650Z)×\left(\mathbb{Z}/650\mathbb{Z}\right)^\times.

nn 2727 301301
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 4.00000i 0.707107i
33 25.3395i 1.62553i 0.582593 + 0.812764i 0.302038π0.302038\pi
−0.582593 + 0.812764i 0.697962π0.697962\pi
44 −16.0000 −0.500000
55 0 0
66 −101.358 −1.14942
77 − 154.564i − 1.19224i −0.802896 0.596120i 0.796708π-0.796708\pi
0.802896 0.596120i 0.203292π-0.203292\pi
88 − 64.0000i − 0.353553i
99 −399.089 −1.64234
1010 0 0
1111 −63.7546 −0.158866 −0.0794328 0.996840i 0.525311π-0.525311\pi
−0.0794328 + 0.996840i 0.525311π0.525311\pi
1212 − 405.431i − 0.812764i
1313 − 169.000i − 0.277350i
1414 618.256 0.843041
1515 0 0
1616 256.000 0.250000
1717 − 374.136i − 0.313984i −0.987600 0.156992i 0.949820π-0.949820\pi
0.987600 0.156992i 0.0501796π-0.0501796\pi
1818 − 1596.35i − 1.16131i
1919 42.3586 0.0269189 0.0134595 0.999909i 0.495716π-0.495716\pi
0.0134595 + 0.999909i 0.495716π0.495716\pi
2020 0 0
2121 3916.57 1.93802
2222 − 255.018i − 0.112335i
2323 − 1131.66i − 0.446062i −0.974811 0.223031i 0.928405π-0.928405\pi
0.974811 0.223031i 0.0715950π-0.0715950\pi
2424 1621.73 0.574711
2525 0 0
2626 676.000 0.196116
2727 − 3955.20i − 1.04414i
2828 2473.03i 0.596120i
2929 −2487.62 −0.549273 −0.274637 0.961548i 0.588558π-0.588558\pi
−0.274637 + 0.961548i 0.588558π0.588558\pi
3030 0 0
3131 9742.11 1.82074 0.910371 0.413792i 0.135796π-0.135796\pi
0.910371 + 0.413792i 0.135796π0.135796\pi
3232 1024.00i 0.176777i
3333 − 1615.51i − 0.258240i
3434 1496.54 0.222020
3535 0 0
3636 6385.42 0.821170
3737 − 5341.14i − 0.641401i −0.947181 0.320700i 0.896082π-0.896082\pi
0.947181 0.320700i 0.103918π-0.103918\pi
3838 169.435i 0.0190346i
3939 4282.37 0.450840
4040 0 0
4141 −7131.82 −0.662584 −0.331292 0.943528i 0.607485π-0.607485\pi
−0.331292 + 0.943528i 0.607485π0.607485\pi
4242 15666.3i 1.37039i
4343 6754.09i 0.557052i 0.960429 + 0.278526i 0.0898459π0.0898459\pi
−0.960429 + 0.278526i 0.910154π0.910154\pi
4444 1020.07 0.0794328
4545 0 0
4646 4526.62 0.315413
4747 14373.8i 0.949132i 0.880220 + 0.474566i 0.157395π0.157395\pi
−0.880220 + 0.474566i 0.842605π0.842605\pi
4848 6486.90i 0.406382i
4949 −7083.06 −0.421435
5050 0 0
5151 9480.40 0.510389
5252 2704.00i 0.138675i
5353 13363.6i 0.653483i 0.945114 + 0.326741i 0.105951π0.105951\pi
−0.945114 + 0.326741i 0.894049π0.894049\pi
5454 15820.8 0.738320
5555 0 0
5656 −9892.10 −0.421520
5757 1073.35i 0.0437575i
5858 − 9950.47i − 0.388395i
5959 −38502.8 −1.44000 −0.720000 0.693974i 0.755858π-0.755858\pi
−0.720000 + 0.693974i 0.755858π0.755858\pi
6060 0 0
6161 28521.0 0.981388 0.490694 0.871332i 0.336743π-0.336743\pi
0.490694 + 0.871332i 0.336743π0.336743\pi
6262 38968.4i 1.28746i
6363 61684.8i 1.95806i
6464 −4096.00 −0.125000
6565 0 0
6666 6462.03 0.182603
6767 43953.6i 1.19621i 0.801418 + 0.598104i 0.204079π0.204079\pi
−0.801418 + 0.598104i 0.795921π0.795921\pi
6868 5986.17i 0.156992i
6969 28675.6 0.725085
7070 0 0
7171 40991.3 0.965040 0.482520 0.875885i 0.339722π-0.339722\pi
0.482520 + 0.875885i 0.339722π0.339722\pi
7272 25541.7i 0.580655i
7373 68431.1i 1.50296i 0.659758 + 0.751478i 0.270659π0.270659\pi
−0.659758 + 0.751478i 0.729341π0.729341\pi
7474 21364.6 0.453539
7575 0 0
7676 −677.738 −0.0134595
7777 9854.17i 0.189406i
7878 17129.5i 0.318792i
7979 −20233.0 −0.364747 −0.182374 0.983229i 0.558378π-0.558378\pi
−0.182374 + 0.983229i 0.558378π0.558378\pi
8080 0 0
8181 3244.21 0.0549409
8282 − 28527.3i − 0.468518i
8383 63147.4i 1.00614i 0.864244 + 0.503072i 0.167797π0.167797\pi
−0.864244 + 0.503072i 0.832203π0.832203\pi
8484 −62665.2 −0.969009
8585 0 0
8686 −27016.4 −0.393895
8787 − 63034.9i − 0.892859i
8888 4080.29i 0.0561674i
8989 116225. 1.55534 0.777668 0.628675i 0.216402π-0.216402\pi
0.777668 + 0.628675i 0.216402π0.216402\pi
9090 0 0
9191 −26121.3 −0.330668
9292 18106.5i 0.223031i
9393 246860.i 2.95967i
9494 −57495.1 −0.671138
9595 0 0
9696 −25947.6 −0.287355
9797 − 59187.1i − 0.638701i −0.947637 0.319351i 0.896535π-0.896535\pi
0.947637 0.319351i 0.103465π-0.103465\pi
9898 − 28332.2i − 0.298000i
9999 25443.7 0.260911
100100 0 0
101101 −10761.5 −0.104971 −0.0524856 0.998622i 0.516714π-0.516714\pi
−0.0524856 + 0.998622i 0.516714π0.516714\pi
102102 37921.6i 0.360899i
103103 94434.4i 0.877076i 0.898713 + 0.438538i 0.144504π0.144504\pi
−0.898713 + 0.438538i 0.855496π0.855496\pi
104104 −10816.0 −0.0980581
105105 0 0
106106 −53454.4 −0.462082
107107 − 81403.2i − 0.687357i −0.939087 0.343678i 0.888327π-0.888327\pi
0.939087 0.343678i 0.111673π-0.111673\pi
108108 63283.3i 0.522071i
109109 145882. 1.17607 0.588037 0.808834i 0.299901π-0.299901\pi
0.588037 + 0.808834i 0.299901π0.299901\pi
110110 0 0
111111 135342. 1.04261
112112 − 39568.4i − 0.298060i
113113 − 86838.6i − 0.639760i −0.947458 0.319880i 0.896357π-0.896357\pi
0.947458 0.319880i 0.103643π-0.103643\pi
114114 −4293.38 −0.0309412
115115 0 0
116116 39801.9 0.274637
117117 67446.0i 0.455503i
118118 − 154011.i − 1.01823i
119119 −57827.9 −0.374344
120120 0 0
121121 −156986. −0.974762
122122 114084.i 0.693946i
123123 − 180717.i − 1.07705i
124124 −155874. −0.910371
125125 0 0
126126 −246739. −1.38456
127127 184919.i 1.01736i 0.860957 + 0.508678i 0.169866π0.169866\pi
−0.860957 + 0.508678i 0.830134π0.830134\pi
128128 − 16384.0i − 0.0883883i
129129 −171145. −0.905504
130130 0 0
131131 −60089.4 −0.305928 −0.152964 0.988232i 0.548882π-0.548882\pi
−0.152964 + 0.988232i 0.548882π0.548882\pi
132132 25848.1i 0.129120i
133133 − 6547.12i − 0.0320938i
134134 −175814. −0.845847
135135 0 0
136136 −23944.7 −0.111010
137137 − 179721.i − 0.818083i −0.912516 0.409042i 0.865863π-0.865863\pi
0.912516 0.409042i 0.134137π-0.134137\pi
138138 114702.i 0.512713i
139139 146077. 0.641276 0.320638 0.947202i 0.396103π-0.396103\pi
0.320638 + 0.947202i 0.396103π0.396103\pi
140140 0 0
141141 −364224. −1.54284
142142 163965.i 0.682386i
143143 10774.5i 0.0440614i
144144 −102167. −0.410585
145145 0 0
146146 −273724. −1.06275
147147 − 179481.i − 0.685055i
148148 85458.2i 0.320700i
149149 295863. 1.09175 0.545877 0.837865i 0.316197π-0.316197\pi
0.545877 + 0.837865i 0.316197π0.316197\pi
150150 0 0
151151 −53761.2 −0.191879 −0.0959394 0.995387i 0.530585π-0.530585\pi
−0.0959394 + 0.995387i 0.530585π0.530585\pi
152152 − 2710.95i − 0.00951728i
153153 149313.i 0.515668i
154154 −39416.7 −0.133930
155155 0 0
156156 −68517.9 −0.225420
157157 − 199374.i − 0.645536i −0.946478 0.322768i 0.895387π-0.895387\pi
0.946478 0.322768i 0.104613π-0.104613\pi
158158 − 80931.9i − 0.257915i
159159 −338627. −1.06225
160160 0 0
161161 −174913. −0.531812
162162 12976.8i 0.0388491i
163163 132038.i 0.389253i 0.980877 + 0.194626i 0.0623494π0.0623494\pi
−0.980877 + 0.194626i 0.937651π0.937651\pi
164164 114109. 0.331292
165165 0 0
166166 −252590. −0.711452
167167 125924.i 0.349396i 0.984622 + 0.174698i 0.0558949π0.0558949\pi
−0.984622 + 0.174698i 0.944105π0.944105\pi
168168 − 250661.i − 0.685193i
169169 −28561.0 −0.0769231
170170 0 0
171171 −16904.9 −0.0442101
172172 − 108065.i − 0.278526i
173173 − 553428.i − 1.40587i −0.711252 0.702937i 0.751872π-0.751872\pi
0.711252 0.702937i 0.248128π-0.248128\pi
174174 252140. 0.631347
175175 0 0
176176 −16321.2 −0.0397164
177177 − 975641.i − 2.34076i
178178 464900.i 1.09979i
179179 471297. 1.09941 0.549707 0.835357i 0.314739π-0.314739\pi
0.549707 + 0.835357i 0.314739π0.314739\pi
180180 0 0
181181 471171. 1.06901 0.534506 0.845165i 0.320498π-0.320498\pi
0.534506 + 0.845165i 0.320498π0.320498\pi
182182 − 104485.i − 0.233817i
183183 722708.i 1.59527i
184184 −72426.0 −0.157707
185185 0 0
186186 −987439. −2.09280
187187 23852.9i 0.0498812i
188188 − 229981.i − 0.474566i
189189 −611332. −1.24487
190190 0 0
191191 319864. 0.634427 0.317214 0.948354i 0.397253π-0.397253\pi
0.317214 + 0.948354i 0.397253π0.397253\pi
192192 − 103790.i − 0.203191i
193193 655179.i 1.26610i 0.774112 + 0.633048i 0.218197π0.218197\pi
−0.774112 + 0.633048i 0.781803π0.781803\pi
194194 236748. 0.451630
195195 0 0
196196 113329. 0.210718
197197 − 680349.i − 1.24901i −0.781021 0.624505i 0.785301π-0.785301\pi
0.781021 0.624505i 0.214699π-0.214699\pi
198198 101775.i 0.184492i
199199 520113. 0.931033 0.465516 0.885039i 0.345869π-0.345869\pi
0.465516 + 0.885039i 0.345869π0.345869\pi
200200 0 0
201201 −1.11376e6 −1.94447
202202 − 43046.1i − 0.0742259i
203203 384496.i 0.654866i
204204 −151686. −0.255194
205205 0 0
206206 −377738. −0.620186
207207 451631.i 0.732585i
208208 − 43264.0i − 0.0693375i
209209 −2700.56 −0.00427649
210210 0 0
211211 121818. 0.188367 0.0941836 0.995555i 0.469976π-0.469976\pi
0.0941836 + 0.995555i 0.469976π0.469976\pi
212212 − 213818.i − 0.326741i
213213 1.03870e6i 1.56870i
214214 325613. 0.486035
215215 0 0
216216 −253133. −0.369160
217217 − 1.50578e6i − 2.17076i
218218 583527.i 0.831609i
219219 −1.73401e6 −2.44310
220220 0 0
221221 −63228.9 −0.0870834
222222 541366.i 0.737240i
223223 826276.i 1.11266i 0.830961 + 0.556331i 0.187791π0.187791\pi
−0.830961 + 0.556331i 0.812209π0.812209\pi
224224 158274. 0.210760
225225 0 0
226226 347355. 0.452378
227227 − 256788.i − 0.330758i −0.986230 0.165379i 0.947115π-0.947115\pi
0.986230 0.165379i 0.0528848π-0.0528848\pi
228228 − 17173.5i − 0.0218787i
229229 −1.19410e6 −1.50470 −0.752351 0.658763i 0.771080π-0.771080\pi
−0.752351 + 0.658763i 0.771080π0.771080\pi
230230 0 0
231231 −249699. −0.307884
232232 159208.i 0.194197i
233233 1.24372e6i 1.50083i 0.660965 + 0.750416i 0.270147π0.270147\pi
−0.660965 + 0.750416i 0.729853π0.729853\pi
234234 −269784. −0.322089
235235 0 0
236236 616045. 0.720000
237237 − 512693.i − 0.592906i
238238 − 231312.i − 0.264701i
239239 −450986. −0.510703 −0.255351 0.966848i 0.582191π-0.582191\pi
−0.255351 + 0.966848i 0.582191π0.582191\pi
240240 0 0
241241 626635. 0.694979 0.347490 0.937684i 0.387034π-0.387034\pi
0.347490 + 0.937684i 0.387034π0.387034\pi
242242 − 627945.i − 0.689261i
243243 − 878908.i − 0.954834i
244244 −456337. −0.490694
245245 0 0
246246 722867. 0.761588
247247 − 7158.61i − 0.00746597i
248248 − 623495.i − 0.643730i
249249 −1.60012e6 −1.63552
250250 0 0
251251 −1.70187e6 −1.70507 −0.852533 0.522674i 0.824934π-0.824934\pi
−0.852533 + 0.522674i 0.824934π0.824934\pi
252252 − 986956.i − 0.979031i
253253 72148.2i 0.0708638i
254254 −739677. −0.719379
255255 0 0
256256 65536.0 0.0625000
257257 − 64954.5i − 0.0613446i −0.999529 0.0306723i 0.990235π-0.990235\pi
0.999529 0.0306723i 0.00976483π-0.00976483\pi
258258 − 684581.i − 0.640288i
259259 −825548. −0.764703
260260 0 0
261261 992780. 0.902094
262262 − 240358.i − 0.216324i
263263 − 263092.i − 0.234541i −0.993100 0.117270i 0.962586π-0.962586\pi
0.993100 0.117270i 0.0374144π-0.0374144\pi
264264 −103392. −0.0913017
265265 0 0
266266 26188.5 0.0226938
267267 2.94508e6i 2.52824i
268268 − 703257.i − 0.598104i
269269 1.13941e6 0.960060 0.480030 0.877252i 0.340626π-0.340626\pi
0.480030 + 0.877252i 0.340626π0.340626\pi
270270 0 0
271271 2.28682e6 1.89151 0.945754 0.324884i 0.105325π-0.105325\pi
0.945754 + 0.324884i 0.105325π0.105325\pi
272272 − 95778.7i − 0.0784959i
273273 − 661901.i − 0.537510i
274274 718884. 0.578472
275275 0 0
276276 −458809. −0.362543
277277 2.43492e6i 1.90671i 0.301851 + 0.953355i 0.402395π0.402395\pi
−0.301851 + 0.953355i 0.597605π0.597605\pi
278278 584308.i 0.453451i
279279 −3.88796e6 −2.99028
280280 0 0
281281 −466127. −0.352159 −0.176079 0.984376i 0.556342π-0.556342\pi
−0.176079 + 0.984376i 0.556342π0.556342\pi
282282 − 1.45690e6i − 1.09095i
283283 838973.i 0.622704i 0.950295 + 0.311352i 0.100782π0.100782\pi
−0.950295 + 0.311352i 0.899218π0.899218\pi
284284 −655860. −0.482520
285285 0 0
286286 −43098.1 −0.0311561
287287 1.10232e6i 0.789959i
288288 − 408667.i − 0.290327i
289289 1.27988e6 0.901414
290290 0 0
291291 1.49977e6 1.03823
292292 − 1.09490e6i − 0.751478i
293293 1.40254e6i 0.954438i 0.878784 + 0.477219i 0.158355π0.158355\pi
−0.878784 + 0.477219i 0.841645π0.841645\pi
294294 717924. 0.484407
295295 0 0
296296 −341833. −0.226769
297297 252162.i 0.165878i
298298 1.18345e6i 0.771986i
299299 −191250. −0.123715
300300 0 0
301301 1.04394e6 0.664140
302302 − 215045.i − 0.135679i
303303 − 272691.i − 0.170634i
304304 10843.8 0.00672974
305305 0 0
306306 −597253. −0.364632
307307 408221.i 0.247200i 0.992332 + 0.123600i 0.0394440π0.0394440\pi
−0.992332 + 0.123600i 0.960556π0.960556\pi
308308 − 157667.i − 0.0947029i
309309 −2.39292e6 −1.42571
310310 0 0
311311 2.26426e6 1.32747 0.663735 0.747968i 0.268970π-0.268970\pi
0.663735 + 0.747968i 0.268970π0.268970\pi
312312 − 274072.i − 0.159396i
313313 2.56466e6i 1.47969i 0.672780 + 0.739843i 0.265100π0.265100\pi
−0.672780 + 0.739843i 0.734900π0.734900\pi
314314 797498. 0.456463
315315 0 0
316316 323727. 0.182374
317317 1.85051e6i 1.03429i 0.855897 + 0.517146i 0.173006π0.173006\pi
−0.855897 + 0.517146i 0.826994π0.826994\pi
318318 − 1.35451e6i − 0.751127i
319319 158597. 0.0872606
320320 0 0
321321 2.06272e6 1.11732
322322 − 699653.i − 0.376048i
323323 − 15847.9i − 0.00845210i
324324 −51907.3 −0.0274705
325325 0 0
326326 −528154. −0.275243
327327 3.69656e6i 1.91174i
328328 456437.i 0.234259i
329329 2.22167e6 1.13159
330330 0 0
331331 −1.94787e6 −0.977215 −0.488607 0.872504i 0.662495π-0.662495\pi
−0.488607 + 0.872504i 0.662495π0.662495\pi
332332 − 1.01036e6i − 0.503072i
333333 2.13159e6i 1.05340i
334334 −503697. −0.247060
335335 0 0
336336 1.00264e6 0.484505
337337 3.42580e6i 1.64319i 0.570073 + 0.821594i 0.306915π0.306915\pi
−0.570073 + 0.821594i 0.693085π0.693085\pi
338338 − 114244.i − 0.0543928i
339339 2.20044e6 1.03995
340340 0 0
341341 −621104. −0.289253
342342 − 67619.4i − 0.0312612i
343343 − 1.50297e6i − 0.689788i
344344 432262. 0.196948
345345 0 0
346346 2.21371e6 0.994103
347347 − 3.01951e6i − 1.34621i −0.739547 0.673105i 0.764960π-0.764960\pi
0.739547 0.673105i 0.235040π-0.235040\pi
348348 1.00856e6i 0.446430i
349349 −1.99404e6 −0.876335 −0.438168 0.898893i 0.644372π-0.644372\pi
−0.438168 + 0.898893i 0.644372π0.644372\pi
350350 0 0
351351 −668429. −0.289593
352352 − 65284.7i − 0.0280837i
353353 − 383180.i − 0.163669i −0.996646 0.0818344i 0.973922π-0.973922\pi
0.996646 0.0818344i 0.0260779π-0.0260779\pi
354354 3.90256e6 1.65517
355355 0 0
356356 −1.85960e6 −0.777668
357357 − 1.46533e6i − 0.608506i
358358 1.88519e6i 0.777404i
359359 3.75111e6 1.53612 0.768058 0.640381i 0.221223π-0.221223\pi
0.768058 + 0.640381i 0.221223π0.221223\pi
360360 0 0
361361 −2.47430e6 −0.999275
362362 1.88469e6i 0.755905i
363363 − 3.97795e6i − 1.58450i
364364 417941. 0.165334
365365 0 0
366366 −2.89083e6 −1.12803
367367 391556.i 0.151750i 0.997117 + 0.0758750i 0.0241750π0.0241750\pi
−0.997117 + 0.0758750i 0.975825π0.975825\pi
368368 − 289704.i − 0.111515i
369369 2.84623e6 1.08819
370370 0 0
371371 2.06553e6 0.779108
372372 − 3.94976e6i − 1.47983i
373373 9831.94i 0.00365904i 0.999998 + 0.00182952i 0.000582355π0.000582355\pi
−0.999998 + 0.00182952i 0.999418π0.999418\pi
374374 −95411.4 −0.0352713
375375 0 0
376376 919922. 0.335569
377377 420407.i 0.152341i
378378 − 2.44533e6i − 0.880254i
379379 −1.87968e6 −0.672180 −0.336090 0.941830i 0.609105π-0.609105\pi
−0.336090 + 0.941830i 0.609105π0.609105\pi
380380 0 0
381381 −4.68576e6 −1.65374
382382 1.27946e6i 0.448608i
383383 − 2.14234e6i − 0.746263i −0.927778 0.373132i 0.878284π-0.878284\pi
0.927778 0.373132i 0.121716π-0.121716\pi
384384 415162. 0.143678
385385 0 0
386386 −2.62072e6 −0.895265
387387 − 2.69548e6i − 0.914869i
388388 946994.i 0.319351i
389389 −1.86897e6 −0.626220 −0.313110 0.949717i 0.601371π-0.601371\pi
−0.313110 + 0.949717i 0.601371π0.601371\pi
390390 0 0
391391 −423393. −0.140056
392392 453316.i 0.149000i
393393 − 1.52263e6i − 0.497295i
394394 2.72139e6 0.883184
395395 0 0
396396 −407100. −0.130456
397397 1.49951e6i 0.477499i 0.971081 + 0.238750i 0.0767375π0.0767375\pi
−0.971081 + 0.238750i 0.923263π0.923263\pi
398398 2.08045e6i 0.658339i
399399 165901. 0.0521694
400400 0 0
401401 2.26959e6 0.704833 0.352416 0.935843i 0.385360π-0.385360\pi
0.352416 + 0.935843i 0.385360π0.385360\pi
402402 − 4.45504e6i − 1.37495i
403403 − 1.64642e6i − 0.504983i
404404 172184. 0.0524856
405405 0 0
406406 −1.53799e6 −0.463060
407407 340522.i 0.101896i
408408 − 606746.i − 0.180450i
409409 −4.88657e6 −1.44443 −0.722214 0.691669i 0.756876π-0.756876\pi
−0.722214 + 0.691669i 0.756876π0.756876\pi
410410 0 0
411411 4.55404e6 1.32982
412412 − 1.51095e6i − 0.438538i
413413 5.95115e6i 1.71682i
414414 −1.80652e6 −0.518016
415415 0 0
416416 173056. 0.0490290
417417 3.70151e6i 1.04241i
418418 − 10802.2i − 0.00302394i
419419 3.07655e6 0.856110 0.428055 0.903753i 0.359199π-0.359199\pi
0.428055 + 0.903753i 0.359199π0.359199\pi
420420 0 0
421421 991694. 0.272692 0.136346 0.990661i 0.456464π-0.456464\pi
0.136346 + 0.990661i 0.456464π0.456464\pi
422422 487272.i 0.133196i
423423 − 5.73642e6i − 1.55880i
424424 855271. 0.231041
425425 0 0
426426 −4.15479e6 −1.10924
427427 − 4.40833e6i − 1.17005i
428428 1.30245e6i 0.343678i
429429 −273021. −0.0716230
430430 0 0
431431 −4.97181e6 −1.28920 −0.644602 0.764519i 0.722977π-0.722977\pi
−0.644602 + 0.764519i 0.722977π0.722977\pi
432432 − 1.01253e6i − 0.261035i
433433 929752.i 0.238313i 0.992875 + 0.119156i 0.0380190π0.0380190\pi
−0.992875 + 0.119156i 0.961981π0.961981\pi
434434 6.02312e6 1.53496
435435 0 0
436436 −2.33411e6 −0.588037
437437 − 47935.4i − 0.0120075i
438438 − 6.93603e6i − 1.72753i
439439 6.24276e6 1.54602 0.773011 0.634393i 0.218750π-0.218750\pi
0.773011 + 0.634393i 0.218750π0.218750\pi
440440 0 0
441441 2.82677e6 0.692140
442442 − 252916.i − 0.0615772i
443443 − 6.12191e6i − 1.48210i −0.671449 0.741051i 0.734328π-0.734328\pi
0.671449 0.741051i 0.265672π-0.265672\pi
444444 −2.16547e6 −0.521307
445445 0 0
446446 −3.30510e6 −0.786771
447447 7.49701e6i 1.77468i
448448 633095.i 0.149030i
449449 −6.81454e6 −1.59522 −0.797611 0.603173i 0.793903π-0.793903\pi
−0.797611 + 0.603173i 0.793903π0.793903\pi
450450 0 0
451451 454686. 0.105262
452452 1.38942e6i 0.319880i
453453 − 1.36228e6i − 0.311904i
454454 1.02715e6 0.233881
455455 0 0
456456 68694.1 0.0154706
457457 1.15462e6i 0.258611i 0.991605 + 0.129306i 0.0412749π0.0412749\pi
−0.991605 + 0.129306i 0.958725π0.958725\pi
458458 − 4.77638e6i − 1.06398i
459459 −1.47978e6 −0.327843
460460 0 0
461461 4.66540e6 1.02244 0.511218 0.859451i 0.329194π-0.329194\pi
0.511218 + 0.859451i 0.329194π0.329194\pi
462462 − 998798.i − 0.217707i
463463 7.33891e6i 1.59103i 0.605932 + 0.795517i 0.292800π0.292800\pi
−0.605932 + 0.795517i 0.707200π0.707200\pi
464464 −636830. −0.137318
465465 0 0
466466 −4.97487e6 −1.06125
467467 1.40701e6i 0.298541i 0.988796 + 0.149270i 0.0476925π0.0476925\pi
−0.988796 + 0.149270i 0.952307π0.952307\pi
468468 − 1.07914e6i − 0.227752i
469469 6.79364e6 1.42617
470470 0 0
471471 5.05204e6 1.04934
472472 2.46418e6i 0.509117i
473473 − 430604.i − 0.0884964i
474474 2.05077e6 0.419248
475475 0 0
476476 925247. 0.187172
477477 − 5.33327e6i − 1.07324i
478478 − 1.80394e6i − 0.361121i
479479 8.02191e6 1.59749 0.798746 0.601668i 0.205497π-0.205497\pi
0.798746 + 0.601668i 0.205497π0.205497\pi
480480 0 0
481481 −902652. −0.177893
482482 2.50654e6i 0.491425i
483483 − 4.43221e6i − 0.864475i
484484 2.51178e6 0.487381
485485 0 0
486486 3.51563e6 0.675169
487487 5.05570e6i 0.965960i 0.875632 + 0.482980i 0.160446π0.160446\pi
−0.875632 + 0.482980i 0.839554π0.839554\pi
488488 − 1.82535e6i − 0.346973i
489489 −3.34579e6 −0.632741
490490 0 0
491491 −7.01316e6 −1.31284 −0.656418 0.754398i 0.727929π-0.727929\pi
−0.656418 + 0.754398i 0.727929π0.727929\pi
492492 2.89147e6i 0.538524i
493493 930706.i 0.172463i
494494 28634.4 0.00527924
495495 0 0
496496 2.49398e6 0.455186
497497 − 6.33578e6i − 1.15056i
498498 − 6.40049e6i − 1.15648i
499499 9.64773e6 1.73450 0.867249 0.497875i 0.165886π-0.165886\pi
0.867249 + 0.497875i 0.165886π0.165886\pi
500500 0 0
501501 −3.19085e6 −0.567953
502502 − 6.80746e6i − 1.20566i
503503 − 9.27398e6i − 1.63435i −0.576386 0.817177i 0.695538π-0.695538\pi
0.576386 0.817177i 0.304462π-0.304462\pi
504504 3.94783e6 0.692280
505505 0 0
506506 −288593. −0.0501083
507507 − 723721.i − 0.125041i
508508 − 2.95871e6i − 0.508678i
509509 1.79545e6 0.307170 0.153585 0.988135i 0.450918π-0.450918\pi
0.153585 + 0.988135i 0.450918π0.450918\pi
510510 0 0
511511 1.05770e7 1.79188
512512 262144.i 0.0441942i
513513 − 167537.i − 0.0281072i
514514 259818. 0.0433772
515515 0 0
516516 2.73832e6 0.452752
517517 − 916395.i − 0.150784i
518518 − 3.30219e6i − 0.540727i
519519 1.40236e7 2.28529
520520 0 0
521521 1.04699e7 1.68986 0.844928 0.534880i 0.179643π-0.179643\pi
0.844928 + 0.534880i 0.179643π0.179643\pi
522522 3.97112e6i 0.637877i
523523 − 239742.i − 0.0383257i −0.999816 0.0191628i 0.993900π-0.993900\pi
0.999816 0.0191628i 0.00610009π-0.00610009\pi
524524 961431. 0.152964
525525 0 0
526526 1.05237e6 0.165845
527527 − 3.64487e6i − 0.571683i
528528 − 413570.i − 0.0645601i
529529 5.15570e6 0.801029
530530 0 0
531531 1.53660e7 2.36497
532532 104754.i 0.0160469i
533533 1.20528e6i 0.183768i
534534 −1.17803e7 −1.78774
535535 0 0
536536 2.81303e6 0.422924
537537 1.19424e7i 1.78713i
538538 4.55763e6i 0.678865i
539539 451578. 0.0669515
540540 0 0
541541 −9.72501e6 −1.42855 −0.714277 0.699863i 0.753244π-0.753244\pi
−0.714277 + 0.699863i 0.753244π0.753244\pi
542542 9.14727e6i 1.33750i
543543 1.19392e7i 1.73771i
544544 383115. 0.0555050
545545 0 0
546546 2.64760e6 0.380077
547547 − 7.17342e6i − 1.02508i −0.858663 0.512540i 0.828705π-0.828705\pi
0.858663 0.512540i 0.171295π-0.171295\pi
548548 2.87554e6i 0.409042i
549549 −1.13824e7 −1.61177
550550 0 0
551551 −105372. −0.0147859
552552 − 1.83524e6i − 0.256356i
553553 3.12729e6i 0.434866i
554554 −9.73967e6 −1.34825
555555 0 0
556556 −2.33723e6 −0.320638
557557 − 4.85134e6i − 0.662558i −0.943533 0.331279i 0.892520π-0.892520\pi
0.943533 0.331279i 0.107480π-0.107480\pi
558558 − 1.55519e7i − 2.11445i
559559 1.14144e6 0.154498
560560 0 0
561561 −604419. −0.0810832
562562 − 1.86451e6i − 0.249014i
563563 2.46602e6i 0.327888i 0.986470 + 0.163944i 0.0524217π0.0524217\pi
−0.986470 + 0.163944i 0.947578π0.947578\pi
564564 5.82759e6 0.771420
565565 0 0
566566 −3.35589e6 −0.440318
567567 − 501438.i − 0.0655028i
568568 − 2.62344e6i − 0.341193i
569569 1.04367e7 1.35139 0.675697 0.737179i 0.263843π-0.263843\pi
0.675697 + 0.737179i 0.263843π0.263843\pi
570570 0 0
571571 4.39937e6 0.564677 0.282338 0.959315i 0.408890π-0.408890\pi
0.282338 + 0.959315i 0.408890π0.408890\pi
572572 − 172392.i − 0.0220307i
573573 8.10518e6i 1.03128i
574574 −4.40930e6 −0.558585
575575 0 0
576576 1.63467e6 0.205293
577577 − 2.03897e6i − 0.254959i −0.991841 0.127480i 0.959311π-0.959311\pi
0.991841 0.127480i 0.0406887π-0.0406887\pi
578578 5.11952e6i 0.637396i
579579 −1.66019e7 −2.05807
580580 0 0
581581 9.76032e6 1.19957
582582 5.99908e6i 0.734137i
583583 − 851991.i − 0.103816i
584584 4.37959e6 0.531376
585585 0 0
586586 −5.61018e6 −0.674890
587587 9.19255e6i 1.10113i 0.834791 + 0.550567i 0.185589π0.185589\pi
−0.834791 + 0.550567i 0.814411π0.814411\pi
588588 2.87170e6i 0.342527i
589589 412662. 0.0490125
590590 0 0
591591 1.72397e7 2.03030
592592 − 1.36733e6i − 0.160350i
593593 7.34348e6i 0.857561i 0.903409 + 0.428780i 0.141057π0.141057\pi
−0.903409 + 0.428780i 0.858943π0.858943\pi
594594 −1.00865e6 −0.117294
595595 0 0
596596 −4.73380e6 −0.545877
597597 1.31794e7i 1.51342i
598598 − 764999.i − 0.0874799i
599599 1.04212e7 1.18672 0.593361 0.804937i 0.297801π-0.297801\pi
0.593361 + 0.804937i 0.297801π0.297801\pi
600600 0 0
601601 7.92846e6 0.895370 0.447685 0.894191i 0.352249π-0.352249\pi
0.447685 + 0.894191i 0.352249π0.352249\pi
602602 4.17576e6i 0.469618i
603603 − 1.75414e7i − 1.96458i
604604 860180. 0.0959394
605605 0 0
606606 1.09077e6 0.120656
607607 6.56441e6i 0.723143i 0.932344 + 0.361572i 0.117760π0.117760\pi
−0.932344 + 0.361572i 0.882240π0.882240\pi
608608 43375.2i 0.00475864i
609609 −9.74293e6 −1.06450
610610 0 0
611611 2.42917e6 0.263242
612612 − 2.38901e6i − 0.257834i
613613 9.93845e6i 1.06824i 0.845410 + 0.534118i 0.179356π0.179356\pi
−0.845410 + 0.534118i 0.820644π0.820644\pi
614614 −1.63288e6 −0.174797
615615 0 0
616616 630667. 0.0669650
617617 1.49986e7i 1.58613i 0.609138 + 0.793064i 0.291516π0.291516\pi
−0.609138 + 0.793064i 0.708484π0.708484\pi
618618 − 9.57167e6i − 1.00813i
619619 −6.67132e6 −0.699818 −0.349909 0.936784i 0.613788π-0.613788\pi
−0.349909 + 0.936784i 0.613788π0.613788\pi
620620 0 0
621621 −4.47593e6 −0.465751
622622 9.05703e6i 0.938663i
623623 − 1.79642e7i − 1.85433i
624624 1.09629e6 0.112710
625625 0 0
626626 −1.02587e7 −1.04630
627627 − 68430.7i − 0.00695156i
628628 3.18999e6i 0.322768i
629629 −1.99831e6 −0.201389
630630 0 0
631631 −1.43883e7 −1.43859 −0.719295 0.694705i 0.755535π-0.755535\pi
−0.719295 + 0.694705i 0.755535π0.755535\pi
632632 1.29491e6i 0.128958i
633633 3.08680e6i 0.306196i
634634 −7.40204e6 −0.731355
635635 0 0
636636 5.41803e6 0.531127
637637 1.19704e6i 0.116885i
638638 634388.i 0.0617026i
639639 −1.63591e7 −1.58492
640640 0 0
641641 −5.54650e6 −0.533180 −0.266590 0.963810i 0.585897π-0.585897\pi
−0.266590 + 0.963810i 0.585897π0.585897\pi
642642 8.25086e6i 0.790063i
643643 − 1.56374e7i − 1.49155i −0.666199 0.745774i 0.732080π-0.732080\pi
0.666199 0.745774i 0.267920π-0.267920\pi
644644 2.79861e6 0.265906
645645 0 0
646646 63391.5 0.00597654
647647 − 9.51432e6i − 0.893546i −0.894647 0.446773i 0.852573π-0.852573\pi
0.894647 0.446773i 0.147427π-0.147427\pi
648648 − 207629.i − 0.0194246i
649649 2.45473e6 0.228766
650650 0 0
651651 3.81557e7 3.52863
652652 − 2.11262e6i − 0.194626i
653653 4.66400e6i 0.428032i 0.976830 + 0.214016i 0.0686544π0.0686544\pi
−0.976830 + 0.214016i 0.931346π0.931346\pi
654654 −1.47863e7 −1.35180
655655 0 0
656656 −1.82575e6 −0.165646
657657 − 2.73101e7i − 2.46837i
658658 8.88669e6i 0.800157i
659659 −3.19305e6 −0.286413 −0.143206 0.989693i 0.545741π-0.545741\pi
−0.143206 + 0.989693i 0.545741π0.545741\pi
660660 0 0
661661 −2.20448e7 −1.96246 −0.981232 0.192829i 0.938234π-0.938234\pi
−0.981232 + 0.192829i 0.938234π0.938234\pi
662662 − 7.79148e6i − 0.690995i
663663 − 1.60219e6i − 0.141556i
664664 4.04143e6 0.355726
665665 0 0
666666 −8.52635e6 −0.744865
667667 2.81513e6i 0.245010i
668668 − 2.01479e6i − 0.174698i
669669 −2.09374e7 −1.80866
670670 0 0
671671 −1.81835e6 −0.155909
672672 4.01057e6i 0.342596i
673673 1.75530e7i 1.49387i 0.664896 + 0.746936i 0.268476π0.268476\pi
−0.664896 + 0.746936i 0.731524π0.731524\pi
674674 −1.37032e7 −1.16191
675675 0 0
676676 456976. 0.0384615
677677 4.12340e6i 0.345767i 0.984942 + 0.172883i 0.0553084π0.0553084\pi
−0.984942 + 0.172883i 0.944692π0.944692\pi
678678 8.80178e6i 0.735354i
679679 −9.14820e6 −0.761485
680680 0 0
681681 6.50688e6 0.537657
682682 − 2.48442e6i − 0.204533i
683683 1.68043e7i 1.37838i 0.724581 + 0.689190i 0.242034π0.242034\pi
−0.724581 + 0.689190i 0.757966π0.757966\pi
684684 270478. 0.0221050
685685 0 0
686686 6.01189e6 0.487754
687687 − 3.02577e7i − 2.44593i
688688 1.72905e6i 0.139263i
689689 2.25845e6 0.181244
690690 0 0
691691 5.42208e6 0.431987 0.215994 0.976395i 0.430701π-0.430701\pi
0.215994 + 0.976395i 0.430701π0.430701\pi
692692 8.85486e6i 0.702937i
693693 − 3.93269e6i − 0.311069i
694694 1.20780e7 0.951915
695695 0 0
696696 −4.03423e6 −0.315673
697697 2.66827e6i 0.208040i
698698 − 7.97616e6i − 0.619662i
699699 −3.15152e7 −2.43965
700700 0 0
701701 −1.78689e7 −1.37342 −0.686710 0.726932i 0.740946π-0.740946\pi
−0.686710 + 0.726932i 0.740946π0.740946\pi
702702 − 2.67372e6i − 0.204773i
703703 − 226243.i − 0.0172658i
704704 261139. 0.0198582
705705 0 0
706706 1.53272e6 0.115731
707707 1.66335e6i 0.125151i
708708 1.56103e7i 1.17038i
709709 7.86353e6 0.587492 0.293746 0.955884i 0.405098π-0.405098\pi
0.293746 + 0.955884i 0.405098π0.405098\pi
710710 0 0
711711 8.07475e6 0.599039
712712 − 7.43840e6i − 0.549894i
713713 − 1.10247e7i − 0.812163i
714714 5.86132e6 0.430279
715715 0 0
716716 −7.54074e6 −0.549707
717717 − 1.14277e7i − 0.830161i
718718 1.50044e7i 1.08620i
719719 1.54686e7 1.11591 0.557956 0.829870i 0.311586π-0.311586\pi
0.557956 + 0.829870i 0.311586π0.311586\pi
720720 0 0
721721 1.45962e7 1.04568
722722 − 9.89722e6i − 0.706594i
723723 1.58786e7i 1.12971i
724724 −7.53874e6 −0.534506
725725 0 0
726726 1.59118e7 1.12041
727727 6.05401e6i 0.424822i 0.977180 + 0.212411i 0.0681316π0.0681316\pi
−0.977180 + 0.212411i 0.931868π0.931868\pi
728728 1.67177e6i 0.116909i
729729 2.30594e7 1.60705
730730 0 0
731731 2.52695e6 0.174905
732732 − 1.15633e7i − 0.797637i
733733 − 1.88061e7i − 1.29282i −0.762989 0.646412i 0.776269π-0.776269\pi
0.762989 0.646412i 0.223731π-0.223731\pi
734734 −1.56622e6 −0.107303
735735 0 0
736736 1.15882e6 0.0788533
737737 − 2.80224e6i − 0.190036i
738738 1.13849e7i 0.769465i
739739 1.12832e7 0.760011 0.380006 0.924984i 0.375922π-0.375922\pi
0.380006 + 0.924984i 0.375922π0.375922\pi
740740 0 0
741741 181395. 0.0121361
742742 8.26214e6i 0.550913i
743743 1.08397e6i 0.0720351i 0.999351 + 0.0360175i 0.0114672π0.0114672\pi
−0.999351 + 0.0360175i 0.988533π0.988533\pi
744744 1.57990e7 1.04640
745745 0 0
746746 −39327.8 −0.00258733
747747 − 2.52014e7i − 1.65243i
748748 − 381646.i − 0.0249406i
749749 −1.25820e7 −0.819494
750750 0 0
751751 −1.03591e7 −0.670230 −0.335115 0.942177i 0.608775π-0.608775\pi
−0.335115 + 0.942177i 0.608775π0.608775\pi
752752 3.67969e6i 0.237283i
753753 − 4.31244e7i − 2.77163i
754754 −1.68163e6 −0.107721
755755 0 0
756756 9.78132e6 0.622433
757757 1.53049e7i 0.970710i 0.874317 + 0.485355i 0.161310π0.161310\pi
−0.874317 + 0.485355i 0.838690π0.838690\pi
758758 − 7.51871e6i − 0.475303i
759759 −1.82820e6 −0.115191
760760 0 0
761761 −2.78390e7 −1.74258 −0.871289 0.490771i 0.836715π-0.836715\pi
−0.871289 + 0.490771i 0.836715π0.836715\pi
762762 − 1.87430e7i − 1.16937i
763763 − 2.25481e7i − 1.40216i
764764 −5.11782e6 −0.317214
765765 0 0
766766 8.56937e6 0.527688
767767 6.50698e6i 0.399384i
768768 1.66065e6i 0.101595i
769769 1.88871e7 1.15173 0.575864 0.817545i 0.304666π-0.304666\pi
0.575864 + 0.817545i 0.304666π0.304666\pi
770770 0 0
771771 1.64591e6 0.0997173
772772 − 1.04829e7i − 0.633048i
773773 1.66647e7i 1.00311i 0.865125 + 0.501556i 0.167239π0.167239\pi
−0.865125 + 0.501556i 0.832761π0.832761\pi
774774 1.07819e7 0.646910
775775 0 0
776776 −3.78798e6 −0.225815
777777 − 2.09190e7i − 1.24305i
778778 − 7.47586e6i − 0.442805i
779779 −302094. −0.0178361
780780 0 0
781781 −2.61338e6 −0.153312
782782 − 1.69357e6i − 0.0990345i
783783 9.83903e6i 0.573519i
784784 −1.81326e6 −0.105359
785785 0 0
786786 6.09054e6 0.351641
787787 − 1.30313e7i − 0.749984i −0.927028 0.374992i 0.877645π-0.877645\pi
0.927028 0.374992i 0.122355π-0.122355\pi
788788 1.08856e7i 0.624505i
789789 6.66661e6 0.381252
790790 0 0
791791 −1.34221e7 −0.762747
792792 − 1.62840e6i − 0.0922460i
793793 − 4.82005e6i − 0.272188i
794794 −5.99803e6 −0.337643
795795 0 0
796796 −8.32180e6 −0.465516
797797 2.29513e6i 0.127986i 0.997950 + 0.0639929i 0.0203835π0.0203835\pi
−0.997950 + 0.0639929i 0.979617π0.979617\pi
798798 663603.i 0.0368893i
799799 5.37775e6 0.298012
800800 0 0
801801 −4.63841e7 −2.55439
802802 9.07835e6i 0.498392i
803803 − 4.36280e6i − 0.238768i
804804 1.78202e7 0.972235
805805 0 0
806806 6.58566e6 0.357077
807807 2.88720e7i 1.56060i
808808 688738.i 0.0371129i
809809 8.57148e6 0.460452 0.230226 0.973137i 0.426053π-0.426053\pi
0.230226 + 0.973137i 0.426053π0.426053\pi
810810 0 0
811811 −3.45545e6 −0.184481 −0.0922407 0.995737i 0.529403π-0.529403\pi
−0.0922407 + 0.995737i 0.529403π0.529403\pi
812812 − 6.15194e6i − 0.327433i
813813 5.79467e7i 3.07470i
814814 −1.36209e6 −0.0720517
815815 0 0
816816 2.42698e6 0.127597
817817 286094.i 0.0149953i
818818 − 1.95463e7i − 1.02137i
819819 1.04247e7 0.543069
820820 0 0
821821 −2.35363e7 −1.21866 −0.609328 0.792918i 0.708561π-0.708561\pi
−0.609328 + 0.792918i 0.708561π0.708561\pi
822822 1.82161e7i 0.940323i
823823 1.60053e7i 0.823693i 0.911253 + 0.411846i 0.135116π0.135116\pi
−0.911253 + 0.411846i 0.864884π0.864884\pi
824824 6.04380e6 0.310093
825825 0 0
826826 −2.38046e7 −1.21398
827827 − 6.11399e6i − 0.310857i −0.987847 0.155429i 0.950324π-0.950324\pi
0.987847 0.155429i 0.0496758π-0.0496758\pi
828828 − 7.22610e6i − 0.366292i
829829 −2.09598e7 −1.05925 −0.529627 0.848231i 0.677668π-0.677668\pi
−0.529627 + 0.848231i 0.677668π0.677668\pi
830830 0 0
831831 −6.16995e7 −3.09941
832832 692224.i 0.0346688i
833833 2.65003e6i 0.132324i
834834 −1.48061e7 −0.737097
835835 0 0
836836 43208.9 0.00213825
837837 − 3.85320e7i − 1.90111i
838838 1.23062e7i 0.605361i
839839 2.39998e7 1.17707 0.588536 0.808471i 0.299704π-0.299704\pi
0.588536 + 0.808471i 0.299704π0.299704\pi
840840 0 0
841841 −1.43229e7 −0.698299
842842 3.96678e6i 0.192822i
843843 − 1.18114e7i − 0.572444i
844844 −1.94909e6 −0.0941836
845845 0 0
846846 2.29457e7 1.10224
847847 2.42645e7i 1.16215i
848848 3.42108e6i 0.163371i
849849 −2.12591e7 −1.01222
850850 0 0
851851 −6.04433e6 −0.286104
852852 − 1.66191e7i − 0.784350i
853853 8.22205e6i 0.386908i 0.981109 + 0.193454i 0.0619690π0.0619690\pi
−0.981109 + 0.193454i 0.938031π0.938031\pi
854854 1.76333e7 0.827350
855855 0 0
856856 −5.20981e6 −0.243017
857857 1.68142e7i 0.782032i 0.920384 + 0.391016i 0.127876π0.127876\pi
−0.920384 + 0.391016i 0.872124π0.872124\pi
858858 − 1.09208e6i − 0.0506451i
859859 3.26713e6 0.151072 0.0755360 0.997143i 0.475933π-0.475933\pi
0.0755360 + 0.997143i 0.475933π0.475933\pi
860860 0 0
861861 −2.79323e7 −1.28410
862862 − 1.98872e7i − 0.911604i
863863 − 3.42932e7i − 1.56741i −0.621136 0.783703i 0.713329π-0.713329\pi
0.621136 0.783703i 0.286671π-0.286671\pi
864864 4.05013e6 0.184580
865865 0 0
866866 −3.71901e6 −0.168513
867867 3.24315e7i 1.46527i
868868 2.40925e7i 1.08538i
869869 1.28994e6 0.0579457
870870 0 0
871871 7.42815e6 0.331769
872872 − 9.33642e6i − 0.415805i
873873 2.36209e7i 1.04896i
874874 191742. 0.00849059
875875 0 0
876876 2.77441e7 1.22155
877877 3.93564e6i 0.172789i 0.996261 + 0.0863946i 0.0275346π0.0275346\pi
−0.996261 + 0.0863946i 0.972465π0.972465\pi
878878 2.49711e7i 1.09320i
879879 −3.55397e7 −1.55147
880880 0 0
881881 −4.25403e7 −1.84655 −0.923274 0.384141i 0.874498π-0.874498\pi
−0.923274 + 0.384141i 0.874498π0.874498\pi
882882 1.13071e7i 0.489417i
883883 1.48855e7i 0.642482i 0.946998 + 0.321241i 0.104100π0.104100\pi
−0.946998 + 0.321241i 0.895900π0.895900\pi
884884 1.01166e6 0.0435417
885885 0 0
886886 2.44877e7 1.04800
887887 3.89095e7i 1.66053i 0.557370 + 0.830264i 0.311811π0.311811\pi
−0.557370 + 0.830264i 0.688189π0.688189\pi
888888 − 8.66186e6i − 0.368620i
889889 2.85819e7 1.21293
890890 0 0
891891 −206833. −0.00872822
892892 − 1.32204e7i − 0.556331i
893893 608854.i 0.0255496i
894894 −2.99880e7 −1.25489
895895 0 0
896896 −2.53238e6 −0.105380
897897 − 4.84617e6i − 0.201102i
898898 − 2.72582e7i − 1.12799i
899899 −2.42346e7 −1.00009
900900 0 0
901901 4.99980e6 0.205183
902902 1.81875e6i 0.0744313i
903903 2.64529e7i 1.07958i
904904 −5.55767e6 −0.226189
905905 0 0
906906 5.44912e6 0.220550
907907 − 4.26154e6i − 0.172008i −0.996295 0.0860039i 0.972590π-0.972590\pi
0.996295 0.0860039i 0.0274098π-0.0274098\pi
908908 4.10861e6i 0.165379i
909909 4.29480e6 0.172399
910910 0 0
911911 −2.52991e7 −1.00997 −0.504985 0.863128i 0.668502π-0.668502\pi
−0.504985 + 0.863128i 0.668502π0.668502\pi
912912 274776.i 0.0109394i
913913 − 4.02594e6i − 0.159842i
914914 −4.61847e6 −0.182866
915915 0 0
916916 1.91055e7 0.752351
917917 9.28767e6i 0.364740i
918918 − 5.91913e6i − 0.231820i
919919 2.20112e7 0.859717 0.429858 0.902896i 0.358563π-0.358563\pi
0.429858 + 0.902896i 0.358563π0.358563\pi
920920 0 0
921921 −1.03441e7 −0.401831
922922 1.86616e7i 0.722972i
923923 − 6.92752e6i − 0.267654i
924924 3.99519e6 0.153942
925925 0 0
926926 −2.93557e7 −1.12503
927927 − 3.76877e7i − 1.44046i
928928 − 2.54732e6i − 0.0970987i
929929 1.80352e6 0.0685615 0.0342808 0.999412i 0.489086π-0.489086\pi
0.0342808 + 0.999412i 0.489086π0.489086\pi
930930 0 0
931931 −300029. −0.0113446
932932 − 1.98995e7i − 0.750416i
933933 5.73751e7i 2.15784i
934934 −5.62803e6 −0.211100
935935 0 0
936936 4.31654e6 0.161045
937937 − 3.06903e7i − 1.14196i −0.820963 0.570981i 0.806563π-0.806563\pi
0.820963 0.570981i 0.193437π-0.193437\pi
938938 2.71746e7i 1.00845i
939939 −6.49872e7 −2.40527
940940 0 0
941941 4.83823e7 1.78120 0.890600 0.454788i 0.150285π-0.150285\pi
0.890600 + 0.454788i 0.150285π0.150285\pi
942942 2.02082e7i 0.741993i
943943 8.07077e6i 0.295553i
944944 −9.85672e6 −0.360000
945945 0 0
946946 1.72242e6 0.0625764
947947 − 3.69874e7i − 1.34023i −0.742259 0.670114i 0.766245π-0.766245\pi
0.742259 0.670114i 0.233755π-0.233755\pi
948948 8.20308e6i 0.296453i
949949 1.15649e7 0.416845
950950 0 0
951951 −4.68910e7 −1.68127
952952 3.70099e6i 0.132350i
953953 − 2.30839e7i − 0.823335i −0.911334 0.411668i 0.864946π-0.864946\pi
0.911334 0.411668i 0.135054π-0.135054\pi
954954 2.13331e7 0.758896
955955 0 0
956956 7.21577e6 0.255351
957957 4.01876e6i 0.141845i
958958 3.20876e7i 1.12960i
959959 −2.77784e7 −0.975351
960960 0 0
961961 6.62795e7 2.31510
962962 − 3.61061e6i − 0.125789i
963963 3.24871e7i 1.12887i
964964 −1.00262e7 −0.347490
965965 0 0
966966 1.77288e7 0.611276
967967 − 2.85570e7i − 0.982079i −0.871137 0.491040i 0.836617π-0.836617\pi
0.871137 0.491040i 0.163383π-0.163383\pi
968968 1.00471e7i 0.344630i
969969 401577. 0.0137391
970970 0 0
971971 −1.99121e7 −0.677748 −0.338874 0.940832i 0.610046π-0.610046\pi
−0.338874 + 0.940832i 0.610046π0.610046\pi
972972 1.40625e7i 0.477417i
973973 − 2.25783e7i − 0.764555i
974974 −2.02228e7 −0.683037
975975 0 0
976976 7.30138e6 0.245347
977977 − 9.96441e6i − 0.333976i −0.985959 0.166988i 0.946596π-0.946596\pi
0.985959 0.166988i 0.0534041π-0.0534041\pi
978978 − 1.33831e7i − 0.447415i
979979 −7.40987e6 −0.247089
980980 0 0
981981 −5.82197e7 −1.93151
982982 − 2.80527e7i − 0.928315i
983983 3.94695e7i 1.30280i 0.758734 + 0.651400i 0.225818π0.225818\pi
−0.758734 + 0.651400i 0.774182π0.774182\pi
984984 −1.15659e7 −0.380794
985985 0 0
986986 −3.72283e6 −0.121950
987987 5.62960e7i 1.83943i
988988 114538.i 0.00373299i
989989 7.64331e6 0.248480
990990 0 0
991991 4.25424e7 1.37606 0.688031 0.725682i 0.258475π-0.258475\pi
0.688031 + 0.725682i 0.258475π0.258475\pi
992992 9.97592e6i 0.321865i
993993 − 4.93580e7i − 1.58849i
994994 2.53431e7 0.813568
995995 0 0
996996 2.56019e7 0.817758
997997 − 1.32492e6i − 0.0422136i −0.999777 0.0211068i 0.993281π-0.993281\pi
0.999777 0.0211068i 0.00671900π-0.00671900\pi
998998 3.85909e7i 1.22647i
999999 −2.11253e7 −0.669713
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 650.6.b.k.599.8 8
5.2 odd 4 650.6.a.l.1.4 4
5.3 odd 4 650.6.a.m.1.1 yes 4
5.4 even 2 inner 650.6.b.k.599.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
650.6.a.l.1.4 4 5.2 odd 4
650.6.a.m.1.1 yes 4 5.3 odd 4
650.6.b.k.599.1 8 5.4 even 2 inner
650.6.b.k.599.8 8 1.1 even 1 trivial