Properties

Label 336.6.q.e.289.1
Level 336336
Weight 66
Character 336.289
Analytic conductor 53.88953.889
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] = [336,6,Mod(193,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 336=2437 336 = 2^{4} \cdot 3 \cdot 7
Weight: k k == 6 6
Character orbit: [χ][\chi] == 336.q (of order 33, degree 22, 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: 53.888963457253.8889634572
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(3,83)\Q(\sqrt{-3}, \sqrt{-83})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x320x221x+441 x^{4} - x^{3} - 20x^{2} - 21x + 441 Copy content Toggle raw display
Coefficient ring: Z[a1,,a19]\Z[a_1, \ldots, a_{19}]
Coefficient ring index: 3 3
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

Embedding invariants

Embedding label 289.1
Root 3.69493+2.71062i-3.69493 + 2.71062i of defining polynomial
Character χ\chi == 336.289
Dual form 336.6.q.e.193.1

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+(4.500007.79423i)q3+(19.3645+33.5404i)q5+(87.5000+95.6596i)q7+(40.5000+70.1481i)q9+(288.195499.168i)q11+391.491q13+348.562q15+(664.850+1151.55i)q17+(471.237816.206i)q19+(351.8421112.46i)q21+(816.040+1413.42i)q23+(812.530+1407.34i)q25+729.000q271463.54q29+(1956.213388.25i)q31+(2593.75+4492.51i)q33+(4902.85+1082.38i)q35+(8150.1714116.5i)q37+(1761.713051.37i)q3913103.8q4114733.5q43+(1568.532716.77i)q45+(3407.26+5901.55i)q47+(1494.50+16740.4i)q49+(5983.6510364.0i)q51+(1005.67+1741.87i)q53+22323.0q558482.26q57+(25726.6+44559.7i)q59+(20548.9+35591.8i)q61+(10254.1+2263.74i)q63+(7581.04+13130.8i)q65+(25289.1+43802.0i)q67+14688.7q6939970.6q71+(27843.3+48226.0i)q73+(7312.7712666.1i)q75+(22533.271245.8i)q77+(31575.7+54690.7i)q79+(3280.505681.99i)q8145572.4q8351498.1q85+(6585.95+11407.2i)q87+(7843.34+13585.1i)q89+(34255.5+37449.9i)q91+(17605.9+30494.3i)q93+(18250.6+31610.9i)q95+3128.49q97+46687.6q99+O(q100)q+(-4.50000 - 7.79423i) q^{3} +(-19.3645 + 33.5404i) q^{5} +(87.5000 + 95.6596i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-288.195 - 499.168i) q^{11} +391.491 q^{13} +348.562 q^{15} +(664.850 + 1151.55i) q^{17} +(471.237 - 816.206i) q^{19} +(351.842 - 1112.46i) q^{21} +(-816.040 + 1413.42i) q^{23} +(812.530 + 1407.34i) q^{25} +729.000 q^{27} -1463.54 q^{29} +(-1956.21 - 3388.25i) q^{31} +(-2593.75 + 4492.51i) q^{33} +(-4902.85 + 1082.38i) q^{35} +(8150.17 - 14116.5i) q^{37} +(-1761.71 - 3051.37i) q^{39} -13103.8 q^{41} -14733.5 q^{43} +(-1568.53 - 2716.77i) q^{45} +(-3407.26 + 5901.55i) q^{47} +(-1494.50 + 16740.4i) q^{49} +(5983.65 - 10364.0i) q^{51} +(1005.67 + 1741.87i) q^{53} +22323.0 q^{55} -8482.26 q^{57} +(25726.6 + 44559.7i) q^{59} +(-20548.9 + 35591.8i) q^{61} +(-10254.1 + 2263.74i) q^{63} +(-7581.04 + 13130.8i) q^{65} +(25289.1 + 43802.0i) q^{67} +14688.7 q^{69} -39970.6 q^{71} +(27843.3 + 48226.0i) q^{73} +(7312.77 - 12666.1i) q^{75} +(22533.2 - 71245.8i) q^{77} +(-31575.7 + 54690.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} -45572.4 q^{83} -51498.1 q^{85} +(6585.95 + 11407.2i) q^{87} +(-7843.34 + 13585.1i) q^{89} +(34255.5 + 37449.9i) q^{91} +(-17605.9 + 30494.3i) q^{93} +(18250.6 + 31610.9i) q^{95} +3128.49 q^{97} +46687.6 q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q18q3+33q5+350q7162q91137q11+1850q13594q15+324q17+2311q191575q21+1596q23395q25+2916q274434q29+4294q3110233q33++184194q99+O(q100) 4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9} - 1137 q^{11} + 1850 q^{13} - 594 q^{15} + 324 q^{17} + 2311 q^{19} - 1575 q^{21} + 1596 q^{23} - 395 q^{25} + 2916 q^{27} - 4434 q^{29} + 4294 q^{31} - 10233 q^{33}+ \cdots + 184194 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/336Z)×\left(\mathbb{Z}/336\mathbb{Z}\right)^\times.

nn 8585 113113 127127 241241
χ(n)\chi(n) 11 11 11 e(13)e\left(\frac{1}{3}\right)

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 0 0
33 −4.50000 7.79423i −0.288675 0.500000i
44 0 0
55 −19.3645 + 33.5404i −0.346403 + 0.599988i −0.985608 0.169049i 0.945930π-0.945930\pi
0.639204 + 0.769037i 0.279264π0.279264\pi
66 0 0
77 87.5000 + 95.6596i 0.674937 + 0.737876i
88 0 0
99 −40.5000 + 70.1481i −0.166667 + 0.288675i
1010 0 0
1111 −288.195 499.168i −0.718133 1.24384i −0.961739 0.273968i 0.911664π-0.911664\pi
0.243606 0.969874i 0.421670π-0.421670\pi
1212 0 0
1313 391.491 0.642486 0.321243 0.946997i 0.395899π-0.395899\pi
0.321243 + 0.946997i 0.395899π0.395899\pi
1414 0 0
1515 348.562 0.399992
1616 0 0
1717 664.850 + 1151.55i 0.557958 + 0.966412i 0.997667 + 0.0682711i 0.0217483π0.0217483\pi
−0.439709 + 0.898140i 0.644918π0.644918\pi
1818 0 0
1919 471.237 816.206i 0.299471 0.518699i −0.676544 0.736402i 0.736523π-0.736523\pi
0.976015 + 0.217703i 0.0698564π0.0698564\pi
2020 0 0
2121 351.842 1112.46i 0.174100 0.550475i
2222 0 0
2323 −816.040 + 1413.42i −0.321656 + 0.557124i −0.980830 0.194866i 0.937573π-0.937573\pi
0.659174 + 0.751991i 0.270906π0.270906\pi
2424 0 0
2525 812.530 + 1407.34i 0.260009 + 0.450350i
2626 0 0
2727 729.000 0.192450
2828 0 0
2929 −1463.54 −0.323155 −0.161577 0.986860i 0.551658π-0.551658\pi
−0.161577 + 0.986860i 0.551658π0.551658\pi
3030 0 0
3131 −1956.21 3388.25i −0.365604 0.633245i 0.623269 0.782008i 0.285804π-0.285804\pi
−0.988873 + 0.148763i 0.952471π0.952471\pi
3232 0 0
3333 −2593.75 + 4492.51i −0.414614 + 0.718133i
3434 0 0
3535 −4902.85 + 1082.38i −0.676517 + 0.149351i
3636 0 0
3737 8150.17 14116.5i 0.978729 1.69521i 0.311691 0.950184i 0.399105π-0.399105\pi
0.667038 0.745024i 0.267562π-0.267562\pi
3838 0 0
3939 −1761.71 3051.37i −0.185470 0.321243i
4040 0 0
4141 −13103.8 −1.21741 −0.608707 0.793395i 0.708312π-0.708312\pi
−0.608707 + 0.793395i 0.708312π0.708312\pi
4242 0 0
4343 −14733.5 −1.21516 −0.607582 0.794257i 0.707860π-0.707860\pi
−0.607582 + 0.794257i 0.707860π0.707860\pi
4444 0 0
4545 −1568.53 2716.77i −0.115468 0.199996i
4646 0 0
4747 −3407.26 + 5901.55i −0.224989 + 0.389692i −0.956316 0.292335i 0.905568π-0.905568\pi
0.731327 + 0.682027i 0.238901π0.238901\pi
4848 0 0
4949 −1494.50 + 16740.4i −0.0889213 + 0.996039i
5050 0 0
5151 5983.65 10364.0i 0.322137 0.557958i
5252 0 0
5353 1005.67 + 1741.87i 0.0491775 + 0.0851779i 0.889566 0.456806i 0.151007π-0.151007\pi
−0.840389 + 0.541984i 0.817673π0.817673\pi
5454 0 0
5555 22323.0 0.995054
5656 0 0
5757 −8482.26 −0.345800
5858 0 0
5959 25726.6 + 44559.7i 0.962170 + 1.66653i 0.717035 + 0.697037i 0.245499π0.245499\pi
0.245135 + 0.969489i 0.421168π0.421168\pi
6060 0 0
6161 −20548.9 + 35591.8i −0.707073 + 1.22469i 0.258865 + 0.965913i 0.416651π0.416651\pi
−0.965938 + 0.258773i 0.916682π0.916682\pi
6262 0 0
6363 −10254.1 + 2263.74i −0.325496 + 0.0718581i
6464 0 0
6565 −7581.04 + 13130.8i −0.222559 + 0.385484i
6666 0 0
6767 25289.1 + 43802.0i 0.688250 + 1.19208i 0.972404 + 0.233305i 0.0749542π0.0749542\pi
−0.284153 + 0.958779i 0.591712π0.591712\pi
6868 0 0
6969 14688.7 0.371416
7070 0 0
7171 −39970.6 −0.941012 −0.470506 0.882397i 0.655929π-0.655929\pi
−0.470506 + 0.882397i 0.655929π0.655929\pi
7272 0 0
7373 27843.3 + 48226.0i 0.611524 + 1.05919i 0.990984 + 0.133983i 0.0427767π0.0427767\pi
−0.379459 + 0.925208i 0.623890π0.623890\pi
7474 0 0
7575 7312.77 12666.1i 0.150117 0.260009i
7676 0 0
7777 22533.2 71245.8i 0.433107 1.36941i
7878 0 0
7979 −31575.7 + 54690.7i −0.569226 + 0.985929i 0.427416 + 0.904055i 0.359424π0.359424\pi
−0.996643 + 0.0818739i 0.973910π0.973910\pi
8080 0 0
8181 −3280.50 5681.99i −0.0555556 0.0962250i
8282 0 0
8383 −45572.4 −0.726116 −0.363058 0.931766i 0.618267π-0.618267\pi
−0.363058 + 0.931766i 0.618267π0.618267\pi
8484 0 0
8585 −51498.1 −0.773114
8686 0 0
8787 6585.95 + 11407.2i 0.0932868 + 0.161577i
8888 0 0
8989 −7843.34 + 13585.1i −0.104961 + 0.181797i −0.913722 0.406340i 0.866805π-0.866805\pi
0.808762 + 0.588137i 0.200138π0.200138\pi
9090 0 0
9191 34255.5 + 37449.9i 0.433637 + 0.474075i
9292 0 0
9393 −17605.9 + 30494.3i −0.211082 + 0.365604i
9494 0 0
9595 18250.6 + 31610.9i 0.207476 + 0.359358i
9696 0 0
9797 3128.49 0.0337603 0.0168801 0.999858i 0.494627π-0.494627\pi
0.0168801 + 0.999858i 0.494627π0.494627\pi
9898 0 0
9999 46687.6 0.478755
100100 0 0
101101 84505.5 + 146368.i 0.824292 + 1.42772i 0.902459 + 0.430776i 0.141760π0.141760\pi
−0.0781663 + 0.996940i 0.524907π0.524907\pi
102102 0 0
103103 56410.1 97705.1i 0.523918 0.907453i −0.475694 0.879611i 0.657803π-0.657803\pi
0.999612 0.0278422i 0.00886360π-0.00886360\pi
104104 0 0
105105 30499.1 + 33343.2i 0.269969 + 0.295144i
106106 0 0
107107 −11154.7 + 19320.6i −0.0941890 + 0.163140i −0.909270 0.416207i 0.863359π-0.863359\pi
0.815081 + 0.579347i 0.196692π0.196692\pi
108108 0 0
109109 41909.8 + 72589.8i 0.337869 + 0.585207i 0.984032 0.177993i 0.0569604π-0.0569604\pi
−0.646162 + 0.763200i 0.723627π0.723627\pi
110110 0 0
111111 −146703. −1.13014
112112 0 0
113113 −40928.4 −0.301529 −0.150764 0.988570i 0.548174π-0.548174\pi
−0.150764 + 0.988570i 0.548174π0.548174\pi
114114 0 0
115115 −31604.4 54740.5i −0.222845 0.385980i
116116 0 0
117117 −15855.4 + 27462.3i −0.107081 + 0.185470i
118118 0 0
119119 −51982.8 + 164360.i −0.336505 + 1.06397i
120120 0 0
121121 −85587.1 + 148241.i −0.531429 + 0.920462i
122122 0 0
123123 58967.2 + 102134.i 0.351437 + 0.608707i
124124 0 0
125125 −183965. −1.05308
126126 0 0
127127 −83270.1 −0.458120 −0.229060 0.973412i 0.573565π-0.573565\pi
−0.229060 + 0.973412i 0.573565π0.573565\pi
128128 0 0
129129 66300.7 + 114836.i 0.350788 + 0.607582i
130130 0 0
131131 −83437.4 + 144518.i −0.424798 + 0.735772i −0.996402 0.0847580i 0.972988π-0.972988\pi
0.571603 + 0.820530i 0.306322π0.306322\pi
132132 0 0
133133 119311. 26339.7i 0.584860 0.129117i
134134 0 0
135135 −14116.7 + 24450.9i −0.0666653 + 0.115468i
136136 0 0
137137 −19111.9 33102.9i −0.0869969 0.150683i 0.819244 0.573446i 0.194394π-0.194394\pi
−0.906240 + 0.422763i 0.861060π0.861060\pi
138138 0 0
139139 −106263. −0.466492 −0.233246 0.972418i 0.574935π-0.574935\pi
−0.233246 + 0.972418i 0.574935π0.574935\pi
140140 0 0
141141 61330.7 0.259795
142142 0 0
143143 −112826. 195420.i −0.461390 0.799151i
144144 0 0
145145 28340.8 49087.8i 0.111942 0.193889i
146146 0 0
147147 137204. 63683.4i 0.523689 0.243071i
148148 0 0
149149 −96277.8 + 166758.i −0.355271 + 0.615348i −0.987164 0.159708i 0.948945π-0.948945\pi
0.631893 + 0.775056i 0.282278π0.282278\pi
150150 0 0
151151 70849.3 + 122715.i 0.252868 + 0.437980i 0.964314 0.264761i 0.0852929π-0.0852929\pi
−0.711447 + 0.702740i 0.751960π0.751960\pi
152152 0 0
153153 −107706. −0.371972
154154 0 0
155155 151524. 0.506586
156156 0 0
157157 −282885. 489972.i −0.915928 1.58643i −0.805536 0.592546i 0.798123π-0.798123\pi
−0.110392 0.993888i 0.535211π-0.535211\pi
158158 0 0
159159 9051.04 15676.9i 0.0283926 0.0491775i
160160 0 0
161161 −206611. + 45612.4i −0.628186 + 0.138682i
162162 0 0
163163 −215101. + 372565.i −0.634121 + 1.09833i 0.352579 + 0.935782i 0.385305π0.385305\pi
−0.986701 + 0.162549i 0.948029π0.948029\pi
164164 0 0
165165 −100454. 173991.i −0.287247 0.497527i
166166 0 0
167167 240265. 0.666653 0.333327 0.942811i 0.391829π-0.391829\pi
0.333327 + 0.942811i 0.391829π0.391829\pi
168168 0 0
169169 −218028. −0.587212
170170 0 0
171171 38170.2 + 66112.7i 0.0998238 + 0.172900i
172172 0 0
173173 89650.1 155279.i 0.227738 0.394454i −0.729399 0.684088i 0.760200π-0.760200\pi
0.957137 + 0.289634i 0.0935337π0.0935337\pi
174174 0 0
175175 −63529.4 + 200869.i −0.156812 + 0.495812i
176176 0 0
177177 231539. 401037.i 0.555509 0.962170i
178178 0 0
179179 −287780. 498449.i −0.671317 1.16275i −0.977531 0.210792i 0.932396π-0.932396\pi
0.306214 0.951963i 0.400938π-0.400938\pi
180180 0 0
181181 581006. 1.31821 0.659105 0.752051i 0.270935π-0.270935\pi
0.659105 + 0.752051i 0.270935π0.270935\pi
182182 0 0
183183 369880. 0.816458
184184 0 0
185185 315648. + 546719.i 0.678070 + 1.17445i
186186 0 0
187187 383213. 663744.i 0.801376 1.38802i
188188 0 0
189189 63787.5 + 69735.8i 0.129892 + 0.142004i
190190 0 0
191191 −330280. + 572062.i −0.655087 + 1.13464i 0.326785 + 0.945099i 0.394035π0.394035\pi
−0.981872 + 0.189545i 0.939299π0.939299\pi
192192 0 0
193193 278655. + 482645.i 0.538485 + 0.932684i 0.998986 + 0.0450243i 0.0143365π0.0143365\pi
−0.460501 + 0.887659i 0.652330π0.652330\pi
194194 0 0
195195 136459. 0.256989
196196 0 0
197197 −761400. −1.39781 −0.698904 0.715216i 0.746328π-0.746328\pi
−0.698904 + 0.715216i 0.746328π0.746328\pi
198198 0 0
199199 −67929.9 117658.i −0.121598 0.210615i 0.798800 0.601597i 0.205469π-0.205469\pi
−0.920398 + 0.390982i 0.872135π0.872135\pi
200200 0 0
201201 227602. 394218.i 0.397361 0.688250i
202202 0 0
203203 −128060. 140002.i −0.218109 0.238448i
204204 0 0
205205 253750. 439507.i 0.421716 0.730434i
206206 0 0
207207 −66099.2 114487.i −0.107219 0.185708i
208208 0 0
209209 −543232. −0.860240
210210 0 0
211211 991157. 1.53263 0.766313 0.642467i 0.222089π-0.222089\pi
0.766313 + 0.642467i 0.222089π0.222089\pi
212212 0 0
213213 179868. + 311540.i 0.271647 + 0.470506i
214214 0 0
215215 285307. 494167.i 0.420937 0.729084i
216216 0 0
217217 152951. 483602.i 0.220496 0.697170i
218218 0 0
219219 250590. 434034.i 0.353064 0.611524i
220220 0 0
221221 260283. + 450823.i 0.358480 + 0.620906i
222222 0 0
223223 −543344. −0.731666 −0.365833 0.930681i 0.619216π-0.619216\pi
−0.365833 + 0.930681i 0.619216π0.619216\pi
224224 0 0
225225 −131630. −0.173340
226226 0 0
227227 8.16704 + 14.1457i 1.05196e−5 + 1.82205e-5i 0.866031 0.499991i 0.166663π-0.166663\pi
−0.866020 + 0.500009i 0.833330π0.833330\pi
228228 0 0
229229 38689.5 67012.2i 0.0487534 0.0844433i −0.840619 0.541627i 0.817808π-0.817808\pi
0.889372 + 0.457184i 0.151142π0.151142\pi
230230 0 0
231231 −656705. + 144978.i −0.809731 + 0.178760i
232232 0 0
233233 51828.6 89769.9i 0.0625432 0.108328i −0.833058 0.553185i 0.813412π-0.813412\pi
0.895602 + 0.444857i 0.146746π0.146746\pi
234234 0 0
235235 −131960. 228561.i −0.155874 0.269981i
236236 0 0
237237 568362. 0.657286
238238 0 0
239239 −689109. −0.780356 −0.390178 0.920739i 0.627587π-0.627587\pi
−0.390178 + 0.920739i 0.627587π0.627587\pi
240240 0 0
241241 −110148. 190782.i −0.122161 0.211590i 0.798458 0.602050i 0.205649π-0.205649\pi
−0.920620 + 0.390460i 0.872316π0.872316\pi
242242 0 0
243243 −29524.5 + 51137.9i −0.0320750 + 0.0555556i
244244 0 0
245245 −532539. 374297.i −0.566809 0.398383i
246246 0 0
247247 184485. 319537.i 0.192406 0.333257i
248248 0 0
249249 205076. + 355201.i 0.209612 + 0.363058i
250250 0 0
251251 −1.43641e6 −1.43912 −0.719558 0.694433i 0.755655π-0.755655\pi
−0.719558 + 0.694433i 0.755655π0.755655\pi
252252 0 0
253253 940714. 0.923966
254254 0 0
255255 231741. + 401388.i 0.223179 + 0.386557i
256256 0 0
257257 454598. 787388.i 0.429334 0.743628i −0.567480 0.823387i 0.692082π-0.692082\pi
0.996814 + 0.0797589i 0.0254151π0.0254151\pi
258258 0 0
259259 2.06352e6 455553.i 1.91143 0.421977i
260260 0 0
261261 59273.5 102665.i 0.0538592 0.0932868i
262262 0 0
263263 374528. + 648702.i 0.333884 + 0.578304i 0.983270 0.182155i 0.0583073π-0.0583073\pi
−0.649386 + 0.760459i 0.724974π0.724974\pi
264264 0 0
265265 −77897.4 −0.0681410
266266 0 0
267267 141180. 0.121198
268268 0 0
269269 −334972. 580189.i −0.282246 0.488865i 0.689691 0.724104i 0.257746π-0.257746\pi
−0.971938 + 0.235238i 0.924413π0.924413\pi
270270 0 0
271271 270330. 468225.i 0.223599 0.387285i −0.732299 0.680983i 0.761553π-0.761553\pi
0.955898 + 0.293698i 0.0948860π0.0948860\pi
272272 0 0
273273 137743. 435519.i 0.111857 0.353672i
274274 0 0
275275 468334. 811178.i 0.373443 0.646821i
276276 0 0
277277 200955. + 348064.i 0.157362 + 0.272558i 0.933916 0.357491i 0.116368π-0.116368\pi
−0.776555 + 0.630050i 0.783035π0.783035\pi
278278 0 0
279279 316906. 0.243736
280280 0 0
281281 −429139. −0.324214 −0.162107 0.986773i 0.551829π-0.551829\pi
−0.162107 + 0.986773i 0.551829π0.551829\pi
282282 0 0
283283 −170463. 295251.i −0.126522 0.219142i 0.795805 0.605553i 0.207048π-0.207048\pi
−0.922327 + 0.386411i 0.873715π0.873715\pi
284284 0 0
285285 164255. 284498.i 0.119786 0.207476i
286286 0 0
287287 −1.14658e6 1.25351e6i −0.821678 0.898301i
288288 0 0
289289 −174123. + 301590.i −0.122634 + 0.212409i
290290 0 0
291291 −14078.2 24384.2i −0.00974575 0.0168801i
292292 0 0
293293 −388847. −0.264612 −0.132306 0.991209i 0.542238π-0.542238\pi
−0.132306 + 0.991209i 0.542238π0.542238\pi
294294 0 0
295295 −1.99273e6 −1.33319
296296 0 0
297297 −210094. 363894.i −0.138205 0.239378i
298298 0 0
299299 −319472. + 553342.i −0.206659 + 0.357945i
300300 0 0
301301 −1.28918e6 1.40940e6i −0.820158 0.896640i
302302 0 0
303303 760549. 1.31731e6i 0.475905 0.824292i
304304 0 0
305305 −795840. 1.37844e6i −0.489865 0.848471i
306306 0 0
307307 2.35747e6 1.42758 0.713789 0.700361i 0.246978π-0.246978\pi
0.713789 + 0.700361i 0.246978π0.246978\pi
308308 0 0
309309 −1.01538e6 −0.604969
310310 0 0
311311 −718314. 1.24416e6i −0.421127 0.729414i 0.574923 0.818208i 0.305032π-0.305032\pi
−0.996050 + 0.0887939i 0.971699π0.971699\pi
312312 0 0
313313 411400. 712566.i 0.237358 0.411116i −0.722598 0.691269i 0.757052π-0.757052\pi
0.959955 + 0.280153i 0.0903853π0.0903853\pi
314314 0 0
315315 122639. 387762.i 0.0696388 0.220186i
316316 0 0
317317 883467. 1.53021e6i 0.493790 0.855269i −0.506184 0.862425i 0.668944π-0.668944\pi
0.999974 + 0.00715584i 0.00227779π0.00227779\pi
318318 0 0
319319 421786. + 730555.i 0.232068 + 0.401954i
320320 0 0
321321 200785. 0.108760
322322 0 0
323323 1.25321e6 0.668370
324324 0 0
325325 318098. + 550962.i 0.167052 + 0.289343i
326326 0 0
327327 377188. 653309.i 0.195069 0.337869i
328328 0 0
329329 −862675. + 190448.i −0.439397 + 0.0970036i
330330 0 0
331331 −1526.14 + 2643.35i −0.000765638 + 0.00132612i −0.866408 0.499337i 0.833577π-0.833577\pi
0.865642 + 0.500663i 0.166910π0.166910\pi
332332 0 0
333333 660164. + 1.14344e6i 0.326243 + 0.565069i
334334 0 0
335335 −1.95885e6 −0.953649
336336 0 0
337337 2.02939e6 0.973398 0.486699 0.873570i 0.338201π-0.338201\pi
0.486699 + 0.873570i 0.338201π0.338201\pi
338338 0 0
339339 184178. + 319006.i 0.0870439 + 0.150764i
340340 0 0
341341 −1.12754e6 + 1.95295e6i −0.525104 + 0.909507i
342342 0 0
343343 −1.73215e6 + 1.32182e6i −0.794969 + 0.606650i
344344 0 0
345345 −284440. + 492665.i −0.128660 + 0.222845i
346346 0 0
347347 1.89109e6 + 3.27547e6i 0.843119 + 1.46033i 0.887245 + 0.461299i 0.152617π0.152617\pi
−0.0441252 + 0.999026i 0.514050π0.514050\pi
348348 0 0
349349 291147. 0.127953 0.0639763 0.997951i 0.479622π-0.479622\pi
0.0639763 + 0.997951i 0.479622π0.479622\pi
350350 0 0
351351 285397. 0.123646
352352 0 0
353353 −192538. 333486.i −0.0822394 0.142443i 0.821972 0.569528i 0.192874π-0.192874\pi
−0.904212 + 0.427085i 0.859541π0.859541\pi
354354 0 0
355355 774013. 1.34063e6i 0.325970 0.564596i
356356 0 0
357357 1.51498e6 334456.i 0.629126 0.138889i
358358 0 0
359359 −1.61507e6 + 2.79738e6i −0.661385 + 1.14555i 0.318866 + 0.947800i 0.396698π0.396698\pi
−0.980252 + 0.197753i 0.936635π0.936635\pi
360360 0 0
361361 793921. + 1.37511e6i 0.320634 + 0.555354i
362362 0 0
363363 1.54057e6 0.613641
364364 0 0
365365 −2.15669e6 −0.847336
366366 0 0
367367 −239779. 415310.i −0.0929280 0.160956i 0.815814 0.578314i 0.196289π-0.196289\pi
−0.908742 + 0.417358i 0.862956π0.862956\pi
368368 0 0
369369 530705. 919208.i 0.202902 0.351437i
370370 0 0
371371 −78630.6 + 248616.i −0.0296590 + 0.0937766i
372372 0 0
373373 −436333. + 755751.i −0.162385 + 0.281259i −0.935724 0.352734i 0.885252π-0.885252\pi
0.773339 + 0.633993i 0.218585π0.218585\pi
374374 0 0
375375 827844. + 1.43387e6i 0.303998 + 0.526540i
376376 0 0
377377 −572965. −0.207622
378378 0 0
379379 2.43493e6 0.870742 0.435371 0.900251i 0.356617π-0.356617\pi
0.435371 + 0.900251i 0.356617π0.356617\pi
380380 0 0
381381 374715. + 649026.i 0.132248 + 0.229060i
382382 0 0
383383 −1.80584e6 + 3.12781e6i −0.629047 + 1.08954i 0.358696 + 0.933454i 0.383222π0.383222\pi
−0.987743 + 0.156087i 0.950112π0.950112\pi
384384 0 0
385385 1.95327e6 + 2.13541e6i 0.671598 + 0.734226i
386386 0 0
387387 596707. 1.03353e6i 0.202527 0.350788i
388388 0 0
389389 87616.2 + 151756.i 0.0293569 + 0.0508477i 0.880331 0.474361i 0.157321π-0.157321\pi
−0.850974 + 0.525208i 0.823987π0.823987\pi
390390 0 0
391391 −2.17018e6 −0.717882
392392 0 0
393393 1.50187e6 0.490515
394394 0 0
395395 −1.22290e6 2.11812e6i −0.394364 0.683058i
396396 0 0
397397 −942577. + 1.63259e6i −0.300152 + 0.519878i −0.976170 0.217007i 0.930371π-0.930371\pi
0.676019 + 0.736885i 0.263704π0.263704\pi
398398 0 0
399399 −742198. 811409.i −0.233393 0.255157i
400400 0 0
401401 −669915. + 1.16033e6i −0.208046 + 0.360346i −0.951099 0.308887i 0.900044π-0.900044\pi
0.743053 + 0.669232i 0.233377π0.233377\pi
402402 0 0
403403 −765839. 1.32647e6i −0.234895 0.406851i
404404 0 0
405405 254101. 0.0769785
406406 0 0
407407 −9.39535e6 −2.81143
408408 0 0
409409 −3.29314e6 5.70388e6i −0.973423 1.68602i −0.685043 0.728503i 0.740217π-0.740217\pi
−0.288381 0.957516i 0.593117π-0.593117\pi
410410 0 0
411411 −172008. + 297926.i −0.0502277 + 0.0869969i
412412 0 0
413413 −2.01149e6 + 6.35996e6i −0.580286 + 1.83476i
414414 0 0
415415 882487. 1.52851e6i 0.251529 0.435661i
416416 0 0
417417 478182. + 828236.i 0.134665 + 0.233246i
418418 0 0
419419 6.96869e6 1.93917 0.969585 0.244754i 0.0787071π-0.0787071\pi
0.969585 + 0.244754i 0.0787071π0.0787071\pi
420420 0 0
421421 3.84041e6 1.05602 0.528010 0.849238i 0.322938π-0.322938\pi
0.528010 + 0.849238i 0.322938π0.322938\pi
422422 0 0
423423 −275988. 478025.i −0.0749962 0.129897i
424424 0 0
425425 −1.08042e6 + 1.87134e6i −0.290149 + 0.502552i
426426 0 0
427427 −5.20272e6 + 1.14858e6i −1.38090 + 0.304854i
428428 0 0
429429 −1.01543e6 + 1.75878e6i −0.266384 + 0.461390i
430430 0 0
431431 1.51818e6 + 2.62957e6i 0.393668 + 0.681854i 0.992930 0.118699i 0.0378725π-0.0378725\pi
−0.599262 + 0.800553i 0.704539π0.704539\pi
432432 0 0
433433 −941529. −0.241332 −0.120666 0.992693i 0.538503π-0.538503\pi
−0.120666 + 0.992693i 0.538503π0.538503\pi
434434 0 0
435435 −510135. −0.129259
436436 0 0
437437 769096. + 1.33211e6i 0.192653 + 0.333686i
438438 0 0
439439 670546. 1.16142e6i 0.166061 0.287626i −0.770971 0.636871i 0.780229π-0.780229\pi
0.937031 + 0.349245i 0.113562π0.113562\pi
440440 0 0
441441 −1.11378e6 782823.i −0.272711 0.191676i
442442 0 0
443443 −386171. + 668867.i −0.0934910 + 0.161931i −0.908978 0.416844i 0.863136π-0.863136\pi
0.815487 + 0.578776i 0.196469π0.196469\pi
444444 0 0
445445 −303765. 526137.i −0.0727174 0.125950i
446446 0 0
447447 1.73300e6 0.410232
448448 0 0
449449 2.25684e6 0.528304 0.264152 0.964481i 0.414908π-0.414908\pi
0.264152 + 0.964481i 0.414908π0.414908\pi
450450 0 0
451451 3.77646e6 + 6.54101e6i 0.874265 + 1.51427i
452452 0 0
453453 637644. 1.10443e6i 0.145993 0.252868i
454454 0 0
455455 −1.91942e6 + 423742.i −0.434653 + 0.0959561i
456456 0 0
457457 −2.14235e6 + 3.71066e6i −0.479844 + 0.831114i −0.999733 0.0231196i 0.992640π-0.992640\pi
0.519889 + 0.854234i 0.325973π0.325973\pi
458458 0 0
459459 484676. + 839483.i 0.107379 + 0.185986i
460460 0 0
461461 −3.10462e6 −0.680387 −0.340193 0.940355i 0.610493π-0.610493\pi
−0.340193 + 0.940355i 0.610493π0.610493\pi
462462 0 0
463463 3.53386e6 0.766121 0.383060 0.923723i 0.374870π-0.374870\pi
0.383060 + 0.923723i 0.374870π0.374870\pi
464464 0 0
465465 −681859. 1.18102e6i −0.146239 0.253293i
466466 0 0
467467 1.36230e6 2.35957e6i 0.289054 0.500657i −0.684530 0.728985i 0.739993π-0.739993\pi
0.973584 + 0.228328i 0.0733258π0.0733258\pi
468468 0 0
469469 −1.97728e6 + 6.25182e6i −0.415085 + 1.31242i
470470 0 0
471471 −2.54597e6 + 4.40975e6i −0.528812 + 0.915928i
472472 0 0
473473 4.24612e6 + 7.35449e6i 0.872649 + 1.51147i
474474 0 0
475475 1.53158e6 0.311461
476476 0 0
477477 −162919. −0.0327850
478478 0 0
479479 489342. + 847566.i 0.0974482 + 0.168785i 0.910628 0.413228i 0.135599π-0.135599\pi
−0.813180 + 0.582013i 0.802265π0.802265\pi
480480 0 0
481481 3.19072e6 5.52649e6i 0.628819 1.08915i
482482 0 0
483483 1.28526e6 + 1.40512e6i 0.250682 + 0.274059i
484484 0 0
485485 −60581.8 + 104931.i −0.0116947 + 0.0202558i
486486 0 0
487487 1.96372e6 + 3.40126e6i 0.375195 + 0.649857i 0.990356 0.138544i 0.0442423π-0.0442423\pi
−0.615161 + 0.788401i 0.710909π0.710909\pi
488488 0 0
489489 3.87181e6 0.732220
490490 0 0
491491 −2.63241e6 −0.492777 −0.246388 0.969171i 0.579244π-0.579244\pi
−0.246388 + 0.969171i 0.579244π0.579244\pi
492492 0 0
493493 −973037. 1.68535e6i −0.180307 0.312301i
494494 0 0
495495 −904083. + 1.56592e6i −0.165842 + 0.287247i
496496 0 0
497497 −3.49743e6 3.82357e6i −0.635123 0.694350i
498498 0 0
499499 1.06272e6 1.84069e6i 0.191059 0.330924i −0.754542 0.656251i 0.772141π-0.772141\pi
0.945601 + 0.325327i 0.105474π0.105474\pi
500500 0 0
501501 −1.08119e6 1.87268e6i −0.192446 0.333327i
502502 0 0
503503 −2.60929e6 −0.459835 −0.229917 0.973210i 0.573846π-0.573846\pi
−0.229917 + 0.973210i 0.573846π0.573846\pi
504504 0 0
505505 −6.54564e6 −1.14215
506506 0 0
507507 981124. + 1.69936e6i 0.169513 + 0.293606i
508508 0 0
509509 5.00911e6 8.67603e6i 0.856970 1.48432i −0.0178348 0.999841i 0.505677π-0.505677\pi
0.874805 0.484475i 0.160989π-0.160989\pi
510510 0 0
511511 −2.17699e6 + 6.88326e6i −0.368811 + 1.16612i
512512 0 0
513513 343532. 595014.i 0.0576333 0.0998238i
514514 0 0
515515 2.18471e6 + 3.78403e6i 0.362974 + 0.628689i
516516 0 0
517517 3.92782e6 0.646287
518518 0 0
519519 −1.61370e6 −0.262969
520520 0 0
521521 2.08970e6 + 3.61947e6i 0.337280 + 0.584186i 0.983920 0.178609i 0.0571598π-0.0571598\pi
−0.646640 + 0.762795i 0.723826π0.723826\pi
522522 0 0
523523 1.80263e6 3.12224e6i 0.288172 0.499128i −0.685202 0.728353i 0.740286π-0.740286\pi
0.973373 + 0.229225i 0.0736193π0.0736193\pi
524524 0 0
525525 1.85150e6 408746.i 0.293174 0.0647225i
526526 0 0
527527 2.60117e6 4.50536e6i 0.407983 0.706648i
528528 0 0
529529 1.88633e6 + 3.26722e6i 0.293075 + 0.507621i
530530 0 0
531531 −4.16770e6 −0.641446
532532 0 0
533533 −5.13003e6 −0.782172
534534 0 0
535535 −432013. 748268.i −0.0652547 0.113025i
536536 0 0
537537 −2.59002e6 + 4.48604e6i −0.387585 + 0.671317i
538538 0 0
539539 8.78699e6 4.07850e6i 1.30277 0.604684i
540540 0 0
541541 3.34083e6 5.78648e6i 0.490751 0.850005i −0.509193 0.860653i 0.670056π-0.670056\pi
0.999943 + 0.0106475i 0.00338926π0.00338926\pi
542542 0 0
543543 −2.61453e6 4.52850e6i −0.380534 0.659105i
544544 0 0
545545 −3.24625e6 −0.468156
546546 0 0
547547 −8.69076e6 −1.24191 −0.620954 0.783847i 0.713255π-0.713255\pi
−0.620954 + 0.783847i 0.713255π0.713255\pi
548548 0 0
549549 −1.66446e6 2.88293e6i −0.235691 0.408229i
550550 0 0
551551 −689676. + 1.19455e6i −0.0967756 + 0.167620i
552552 0 0
553553 −7.99456e6 + 1.76492e6i −1.11168 + 0.245421i
554554 0 0
555555 2.84084e6 4.92047e6i 0.391484 0.678070i
556556 0 0
557557 −3.12371e6 5.41042e6i −0.426612 0.738913i 0.569958 0.821674i 0.306960π-0.306960\pi
−0.996569 + 0.0827611i 0.973626π0.973626\pi
558558 0 0
559559 −5.76803e6 −0.780725
560560 0 0
561561 −6.89783e6 −0.925349
562562 0 0
563563 −5.95223e6 1.03096e7i −0.791423 1.37078i −0.925086 0.379758i 0.876007π-0.876007\pi
0.133663 0.991027i 0.457326π-0.457326\pi
564564 0 0
565565 792560. 1.37275e6i 0.104451 0.180914i
566566 0 0
567567 256493. 810986.i 0.0335057 0.105939i
568568 0 0
569569 −10707.2 + 18545.4i −0.00138642 + 0.00240135i −0.866718 0.498799i 0.833775π-0.833775\pi
0.865331 + 0.501200i 0.167108π0.167108\pi
570570 0 0
571571 3.55823e6 + 6.16304e6i 0.456714 + 0.791051i 0.998785 0.0492811i 0.0156930π-0.0156930\pi
−0.542071 + 0.840333i 0.682360π0.682360\pi
572572 0 0
573573 5.94504e6 0.756429
574574 0 0
575575 −2.65223e6 −0.334534
576576 0 0
577577 5.33259e6 + 9.23632e6i 0.666805 + 1.15494i 0.978793 + 0.204854i 0.0656719π0.0656719\pi
−0.311988 + 0.950086i 0.600995π0.600995\pi
578578 0 0
579579 2.50790e6 4.34380e6i 0.310895 0.538485i
580580 0 0
581581 −3.98758e6 4.35943e6i −0.490083 0.535784i
582582 0 0
583583 579659. 1.00400e6i 0.0706319 0.122338i
584584 0 0
585585 −614065. 1.06359e6i −0.0741864 0.128495i
586586 0 0
587587 1.30101e7 1.55843 0.779213 0.626759i 0.215619π-0.215619\pi
0.779213 + 0.626759i 0.215619π0.215619\pi
588588 0 0
589589 −3.68735e6 −0.437952
590590 0 0
591591 3.42630e6 + 5.93453e6i 0.403512 + 0.698904i
592592 0 0
593593 −2.13043e6 + 3.69002e6i −0.248789 + 0.430915i −0.963190 0.268821i 0.913366π-0.913366\pi
0.714401 + 0.699736i 0.246699π0.246699\pi
594594 0 0
595595 −4.50608e6 4.92628e6i −0.521803 0.570462i
596596 0 0
597597 −611369. + 1.05892e6i −0.0702049 + 0.121598i
598598 0 0
599599 −6.89790e6 1.19475e7i −0.785507 1.36054i −0.928696 0.370842i 0.879069π-0.879069\pi
0.143189 0.989695i 0.454264π-0.454264\pi
600600 0 0
601601 4.99695e6 0.564311 0.282155 0.959369i 0.408951π-0.408951\pi
0.282155 + 0.959369i 0.408951π0.408951\pi
602602 0 0
603603 −4.09683e6 −0.458833
604604 0 0
605605 −3.31471e6 5.74125e6i −0.368177 0.637702i
606606 0 0
607607 −1.52473e6 + 2.64091e6i −0.167966 + 0.290926i −0.937705 0.347434i 0.887053π-0.887053\pi
0.769739 + 0.638359i 0.220387π0.220387\pi
608608 0 0
609609 −514937. + 1.62814e6i −0.0562614 + 0.177889i
610610 0 0
611611 −1.33391e6 + 2.31040e6i −0.144552 + 0.250372i
612612 0 0
613613 3.51813e6 + 6.09357e6i 0.378147 + 0.654969i 0.990793 0.135388i 0.0432282π-0.0432282\pi
−0.612646 + 0.790357i 0.709895π0.709895\pi
614614 0 0
615615 −4.56749e6 −0.486956
616616 0 0
617617 1.00066e7 1.05822 0.529108 0.848554i 0.322527π-0.322527\pi
0.529108 + 0.848554i 0.322527π0.322527\pi
618618 0 0
619619 −3.27533e6 5.67304e6i −0.343581 0.595099i 0.641514 0.767111i 0.278307π-0.278307\pi
−0.985095 + 0.172012i 0.944973π0.944973\pi
620620 0 0
621621 −594893. + 1.03038e6i −0.0619027 + 0.107219i
622622 0 0
623623 −1.98583e6 + 438403.i −0.204985 + 0.0452536i
624624 0 0
625625 1.02325e6 1.77232e6i 0.104781 0.181485i
626626 0 0
627627 2.44455e6 + 4.23408e6i 0.248330 + 0.430120i
628628 0 0
629629 2.16746e7 2.18436
630630 0 0
631631 −2.22672e6 −0.222635 −0.111317 0.993785i 0.535507π-0.535507\pi
−0.111317 + 0.993785i 0.535507π0.535507\pi
632632 0 0
633633 −4.46021e6 7.72531e6i −0.442431 0.766313i
634634 0 0
635635 1.61249e6 2.79291e6i 0.158694 0.274867i
636636 0 0
637637 −585084. + 6.55373e6i −0.0571307 + 0.639941i
638638 0 0
639639 1.61881e6 2.80386e6i 0.156835 0.271647i
640640 0 0
641641 7.96698e6 + 1.37992e7i 0.765859 + 1.32651i 0.939791 + 0.341749i 0.111019π0.111019\pi
−0.173932 + 0.984758i 0.555647π0.555647\pi
642642 0 0
643643 1.49933e7 1.43011 0.715056 0.699067i 0.246401π-0.246401\pi
0.715056 + 0.699067i 0.246401π0.246401\pi
644644 0 0
645645 −5.13553e6 −0.486056
646646 0 0
647647 6.49027e6 + 1.12415e7i 0.609540 + 1.05575i 0.991316 + 0.131500i 0.0419792π0.0419792\pi
−0.381776 + 0.924255i 0.624687π0.624687\pi
648648 0 0
649649 1.48285e7 2.56838e7i 1.38193 2.39357i
650650 0 0
651651 −4.45758e6 + 984079.i −0.412237 + 0.0910075i
652652 0 0
653653 8.27460e6 1.43320e7i 0.759389 1.31530i −0.183774 0.982969i 0.558831π-0.558831\pi
0.943163 0.332332i 0.107835π-0.107835\pi
654654 0 0
655655 −3.23145e6 5.59704e6i −0.294303 0.509748i
656656 0 0
657657 −4.51062e6 −0.407683
658658 0 0
659659 −5.86879e6 −0.526423 −0.263212 0.964738i 0.584782π-0.584782\pi
−0.263212 + 0.964738i 0.584782π0.584782\pi
660660 0 0
661661 −3.63843e6 6.30195e6i −0.323900 0.561011i 0.657389 0.753551i 0.271661π-0.271661\pi
−0.981289 + 0.192540i 0.938327π0.938327\pi
662662 0 0
663663 2.34255e6 4.05741e6i 0.206969 0.358480i
664664 0 0
665665 −1.42696e6 + 4.51179e6i −0.125129 + 0.395635i
666666 0 0
667667 1.19431e6 2.06861e6i 0.103945 0.180038i
668668 0 0
669669 2.44505e6 + 4.23495e6i 0.211214 + 0.365833i
670670 0 0
671671 2.36884e7 2.03109
672672 0 0
673673 −1.82417e7 −1.55248 −0.776241 0.630437i 0.782876π-0.782876\pi
−0.776241 + 0.630437i 0.782876π0.782876\pi
674674 0 0
675675 592334. + 1.02595e6i 0.0500388 + 0.0866698i
676676 0 0
677677 3.88203e6 6.72387e6i 0.325527 0.563829i −0.656092 0.754681i 0.727792π-0.727792\pi
0.981619 + 0.190852i 0.0611250π0.0611250\pi
678678 0 0
679679 273743. + 299270.i 0.0227860 + 0.0249109i
680680 0 0
681681 73.5034 127.312i 6.07351e−6 1.05196e-5i
682682 0 0
683683 4.28162e6 + 7.41598e6i 0.351201 + 0.608299i 0.986460 0.164001i 0.0524399π-0.0524399\pi
−0.635259 + 0.772299i 0.719107π0.719107\pi
684684 0 0
685685 1.48038e6 0.120544
686686 0 0
687687 −696411. −0.0562955
688688 0 0
689689 393712. + 681928.i 0.0315959 + 0.0547256i
690690 0 0
691691 8.22547e6 1.42469e7i 0.655338 1.13508i −0.326471 0.945207i 0.605860π-0.605860\pi
0.981809 0.189872i 0.0608071π-0.0608071\pi
692692 0 0
693693 4.08516e6 + 4.46611e6i 0.323129 + 0.353262i
694694 0 0
695695 2.05773e6 3.56409e6i 0.161594 0.279890i
696696 0 0
697697 −8.71208e6 1.50898e7i −0.679266 1.17652i
698698 0 0
699699 −932916. −0.0722187
700700 0 0
701701 −1.66928e7 −1.28302 −0.641512 0.767113i 0.721693π-0.721693\pi
−0.641512 + 0.767113i 0.721693π0.721693\pi
702702 0 0
703703 −7.68132e6 1.33044e7i −0.586202 1.01533i
704704 0 0
705705 −1.18764e6 + 2.05705e6i −0.0899937 + 0.155874i
706706 0 0
707707 −6.60725e6 + 2.08909e7i −0.497132 + 1.57184i
708708 0 0
709709 −2.80890e6 + 4.86515e6i −0.209855 + 0.363480i −0.951669 0.307126i 0.900633π-0.900633\pi
0.741813 + 0.670606i 0.233966π0.233966\pi
710710 0 0
711711 −2.55763e6 4.42994e6i −0.189742 0.328643i
712712 0 0
713713 6.38537e6 0.470395
714714 0 0
715715 8.73927e6 0.639308
716716 0 0
717717 3.10099e6 + 5.37107e6i 0.225269 + 0.390178i
718718 0 0
719719 5.18592e6 8.98227e6i 0.374113 0.647983i −0.616081 0.787683i 0.711280π-0.711280\pi
0.990194 + 0.139700i 0.0446138π0.0446138\pi
720720 0 0
721721 1.42823e7 3.15303e6i 1.02320 0.225887i
722722 0 0
723723 −991332. + 1.71704e6i −0.0705299 + 0.122161i
724724 0 0
725725 −1.18917e6 2.05971e6i −0.0840233 0.145533i
726726 0 0
727727 −1.15369e7 −0.809565 −0.404783 0.914413i 0.632653π-0.632653\pi
−0.404783 + 0.914413i 0.632653π0.632653\pi
728728 0 0
729729 531441. 0.0370370
730730 0 0
731731 −9.79557e6 1.69664e7i −0.678010 1.17435i
732732 0 0
733733 −7.47349e6 + 1.29445e7i −0.513764 + 0.889865i 0.486109 + 0.873898i 0.338416π0.338416\pi
−0.999873 + 0.0159667i 0.994917π0.994917\pi
734734 0 0
735735 −520925. + 5.83507e6i −0.0355678 + 0.398408i
736736 0 0
737737 1.45764e7 2.52470e7i 0.988510 1.71215i
738738 0 0
739739 4.50682e6 + 7.80604e6i 0.303570 + 0.525799i 0.976942 0.213505i 0.0684880π-0.0684880\pi
−0.673372 + 0.739304i 0.735155π0.735155\pi
740740 0 0
741741 −3.32073e6 −0.222171
742742 0 0
743743 2.10239e7 1.39714 0.698571 0.715541i 0.253820π-0.253820\pi
0.698571 + 0.715541i 0.253820π0.253820\pi
744744 0 0
745745 −3.72875e6 6.45838e6i −0.246134 0.426317i
746746 0 0
747747 1.84568e6 3.19681e6i 0.121019 0.209612i
748748 0 0
749749 −2.82424e6 + 623493.i −0.183949 + 0.0406094i
750750 0 0
751751 2.02110e6 3.50064e6i 0.130764 0.226489i −0.793207 0.608952i 0.791590π-0.791590\pi
0.923971 + 0.382462i 0.124924π0.124924\pi
752752 0 0
753753 6.46387e6 + 1.11957e7i 0.415437 + 0.719558i
754754 0 0
755755 −5.48786e6 −0.350377
756756 0 0
757757 −1.82059e7 −1.15471 −0.577353 0.816495i 0.695914π-0.695914\pi
−0.577353 + 0.816495i 0.695914π0.695914\pi
758758 0 0
759759 −4.23321e6 7.33214e6i −0.266726 0.461983i
760760 0 0
761761 9.45999e6 1.63852e7i 0.592146 1.02563i −0.401797 0.915729i 0.631614π-0.631614\pi
0.993943 0.109898i 0.0350526π-0.0350526\pi
762762 0 0
763763 −3.27681e6 + 1.03607e7i −0.203770 + 0.644283i
764764 0 0
765765 2.08567e6 3.61249e6i 0.128852 0.223179i
766766 0 0
767767 1.00717e7 + 1.74447e7i 0.618180 + 1.07072i
768768 0 0
769769 −1.12831e7 −0.688037 −0.344019 0.938963i 0.611788π-0.611788\pi
−0.344019 + 0.938963i 0.611788π0.611788\pi
770770 0 0
771771 −8.18277e6 −0.495752
772772 0 0
773773 1.84282e6 + 3.19186e6i 0.110926 + 0.192130i 0.916144 0.400849i 0.131285π-0.131285\pi
−0.805218 + 0.592979i 0.797952π0.797952\pi
774774 0 0
775775 3.17896e6 5.50611e6i 0.190121 0.329299i
776776 0 0
777777 −1.28365e7 1.40335e7i −0.762772 0.833902i
778778 0 0
779779 −6.17501e6 + 1.06954e7i −0.364581 + 0.631472i
780780 0 0
781781 1.15193e7 + 1.99521e7i 0.675771 + 1.17047i
782782 0 0
783783 −1.06692e6 −0.0621912
784784 0 0
785785 2.19118e7 1.26912
786786 0 0
787787 −7.03741e6 1.21891e7i −0.405019 0.701514i 0.589304 0.807911i 0.299402π-0.299402\pi
−0.994324 + 0.106397i 0.966069π0.966069\pi
788788 0 0
789789 3.37076e6 5.83832e6i 0.192768 0.333884i
790790 0 0
791791 −3.58124e6 3.91519e6i −0.203513 0.222491i
792792 0 0
793793 −8.04472e6 + 1.39339e7i −0.454284 + 0.786844i
794794 0 0
795795 350538. + 607150.i 0.0196706 + 0.0340705i
796796 0 0
797797 −1.75191e7 −0.976937 −0.488469 0.872582i 0.662444π-0.662444\pi
−0.488469 + 0.872582i 0.662444π0.662444\pi
798798 0 0
799799 −9.06127e6 −0.502137
800800 0 0
801801 −635311. 1.10039e6i −0.0349869 0.0605990i
802802 0 0
803803 1.60486e7 2.77970e7i 0.878311 1.52128i
804804 0 0
805805 2.47106e6 7.81306e6i 0.134398 0.424944i
806806 0 0
807807 −3.01475e6 + 5.22170e6i −0.162955 + 0.282246i
808808 0 0
809809 −407261. 705396.i −0.0218777 0.0378932i 0.854879 0.518827i 0.173631π-0.173631\pi
−0.876757 + 0.480934i 0.840298π0.840298\pi
810810 0 0
811811 −1.26533e7 −0.675540 −0.337770 0.941229i 0.609673π-0.609673\pi
−0.337770 + 0.941229i 0.609673π0.609673\pi
812812 0 0
813813 −4.86593e6 −0.258190
814814 0 0
815815 −8.33064e6 1.44291e7i −0.439324 0.760931i
816816 0 0
817817 −6.94297e6 + 1.20256e7i −0.363907 + 0.630305i
818818 0 0
819819 −4.01438e6 + 886236.i −0.209126 + 0.0461678i
820820 0 0
821821 −2.13697e6 + 3.70133e6i −0.110647 + 0.191646i −0.916031 0.401107i 0.868626π-0.868626\pi
0.805384 + 0.592753i 0.201959π0.201959\pi
822822 0 0
823823 8.52317e6 + 1.47626e7i 0.438633 + 0.759735i 0.997584 0.0694656i 0.0221294π-0.0221294\pi
−0.558951 + 0.829201i 0.688796π0.688796\pi
824824 0 0
825825 −8.43001e6 −0.431214
826826 0 0
827827 −2.60828e6 −0.132614 −0.0663071 0.997799i 0.521122π-0.521122\pi
−0.0663071 + 0.997799i 0.521122π0.521122\pi
828828 0 0
829829 −1.06932e7 1.85212e7i −0.540409 0.936016i −0.998880 0.0473066i 0.984936π-0.984936\pi
0.458471 0.888709i 0.348397π-0.348397\pi
830830 0 0
831831 1.80859e6 3.13257e6i 0.0908528 0.157362i
832832 0 0
833833 −2.02711e7 + 9.40887e6i −1.01220 + 0.469813i
834834 0 0
835835 −4.65263e6 + 8.05859e6i −0.230931 + 0.399984i
836836 0 0
837837 −1.42608e6 2.47004e6i −0.0703605 0.121868i
838838 0 0
839839 −771393. −0.0378330 −0.0189165 0.999821i 0.506022π-0.506022\pi
−0.0189165 + 0.999821i 0.506022π0.506022\pi
840840 0 0
841841 −1.83692e7 −0.895571
842842 0 0
843843 1.93112e6 + 3.34480e6i 0.0935925 + 0.162107i
844844 0 0
845845 4.22200e6 7.31272e6i 0.203412 0.352320i
846846 0 0
847847 −2.16696e7 + 4.78388e6i −1.03787 + 0.229125i
848848 0 0
849849 −1.53417e6 + 2.65726e6i −0.0730474 + 0.126522i
850850 0 0
851851 1.33017e7 + 2.30393e7i 0.629628 + 1.09055i
852852 0 0
853853 2.94032e7 1.38364 0.691818 0.722072i 0.256810π-0.256810\pi
0.691818 + 0.722072i 0.256810π0.256810\pi
854854 0 0
855855 −2.95659e6 −0.138317
856856 0 0
857857 1.32505e7 + 2.29505e7i 0.616283 + 1.06743i 0.990158 + 0.139954i 0.0446955π0.0446955\pi
−0.373875 + 0.927479i 0.621971π0.621971\pi
858858 0 0
859859 −1.83759e7 + 3.18281e7i −0.849702 + 1.47173i 0.0317727 + 0.999495i 0.489885π0.489885\pi
−0.881475 + 0.472232i 0.843449π0.843449\pi
860860 0 0
861861 −4.61048e6 + 1.45775e7i −0.211952 + 0.670156i
862862 0 0
863863 5.90946e6 1.02355e7i 0.270098 0.467823i −0.698789 0.715328i 0.746277π-0.746277\pi
0.968887 + 0.247505i 0.0796106π0.0796106\pi
864864 0 0
865865 3.47207e6 + 6.01379e6i 0.157778 + 0.273280i
866866 0 0
867867 3.13421e6 0.141606
868868 0 0
869869 3.63998e7 1.63512
870870 0 0
871871 9.90046e6 + 1.71481e7i 0.442191 + 0.765897i
872872 0 0
873873 −126704. + 219458.i −0.00562671 + 0.00974575i
874874 0 0
875875 −1.60970e7 1.75980e7i −0.710762 0.777042i
876876 0 0
877877 −3.96754e6 + 6.87199e6i −0.174190 + 0.301706i −0.939881 0.341503i 0.889064π-0.889064\pi
0.765691 + 0.643209i 0.222397π0.222397\pi
878878 0 0
879879 1.74981e6 + 3.03077e6i 0.0763870 + 0.132306i
880880 0 0
881881 4.20152e7 1.82375 0.911877 0.410464i 0.134633π-0.134633\pi
0.911877 + 0.410464i 0.134633π0.134633\pi
882882 0 0
883883 −2.12461e7 −0.917016 −0.458508 0.888690i 0.651616π-0.651616\pi
−0.458508 + 0.888690i 0.651616π0.651616\pi
884884 0 0
885885 8.96729e6 + 1.55318e7i 0.384860 + 0.666597i
886886 0 0
887887 9.42452e6 1.63238e7i 0.402208 0.696644i −0.591784 0.806096i 0.701576π-0.701576\pi
0.993992 + 0.109452i 0.0349097π0.0349097\pi
888888 0 0
889889 −7.28613e6 7.96558e6i −0.309202 0.338036i
890890 0 0
891891 −1.89085e6 + 3.27504e6i −0.0797925 + 0.138205i
892892 0 0
893893 3.21125e6 + 5.56205e6i 0.134755 + 0.233403i
894894 0 0
895895 2.22909e7 0.930186
896896 0 0
897897 5.75050e6 0.238630
898898 0 0
899899 2.86300e6 + 4.95886e6i 0.118147 + 0.204636i
900900 0 0
901901 −1.33724e6 + 2.31617e6i −0.0548780 + 0.0950514i
902902 0 0
903903 −5.18387e6 + 1.63905e7i −0.211561 + 0.668917i
904904 0 0
905905 −1.12509e7 + 1.94872e7i −0.456632 + 0.790910i
906906 0 0
907907 −3.09723e6 5.36456e6i −0.125013 0.216529i 0.796725 0.604342i 0.206564π-0.206564\pi
−0.921738 + 0.387813i 0.873231π0.873231\pi
908908 0 0
909909 −1.36899e7 −0.549528
910910 0 0
911911 2.50171e7 0.998712 0.499356 0.866397i 0.333570π-0.333570\pi
0.499356 + 0.866397i 0.333570π0.333570\pi
912912 0 0
913913 1.31337e7 + 2.27483e7i 0.521448 + 0.903174i
914914 0 0
915915 −7.16256e6 + 1.24059e7i −0.282824 + 0.489865i
916916 0 0
917917 −2.11253e7 + 4.66373e6i −0.829620 + 0.183151i
918918 0 0
919919 7.74961e6 1.34227e7i 0.302685 0.524266i −0.674058 0.738678i 0.735450π-0.735450\pi
0.976743 + 0.214412i 0.0687836π0.0687836\pi
920920 0 0
921921 −1.06086e7 1.83746e7i −0.412106 0.713789i
922922 0 0
923923 −1.56481e7 −0.604587
924924 0 0
925925 2.64890e7 1.01791
926926 0 0
927927 4.56921e6 + 7.91411e6i 0.174639 + 0.302484i
928928 0 0
929929 −1.87643e7 + 3.25007e7i −0.713333 + 1.23553i 0.250266 + 0.968177i 0.419482π0.419482\pi
−0.963599 + 0.267352i 0.913852π0.913852\pi
930930 0 0
931931 1.29594e7 + 9.10852e6i 0.490015 + 0.344408i
932932 0 0
933933 −6.46483e6 + 1.11974e7i −0.243138 + 0.421127i
934934 0 0
935935 1.48415e7 + 2.57062e7i 0.555198 + 0.961632i
936936 0 0
937937 −1.08298e7 −0.402969 −0.201485 0.979492i 0.564577π-0.564577\pi
−0.201485 + 0.979492i 0.564577π0.564577\pi
938938 0 0
939939 −7.40520e6 −0.274077
940940 0 0
941941 −1.62295e7 2.81104e7i −0.597492 1.03489i −0.993190 0.116505i 0.962831π-0.962831\pi
0.395699 0.918380i 0.370503π-0.370503\pi
942942 0 0
943943 1.06932e7 1.85212e7i 0.391589 0.678251i
944944 0 0
945945 −3.57418e6 + 789054.i −0.130196 + 0.0287427i
946946 0 0
947947 3.76939e6 6.52877e6i 0.136583 0.236568i −0.789618 0.613598i 0.789721π-0.789721\pi
0.926201 + 0.377030i 0.123055π0.123055\pi
948948 0 0
949949 1.09004e7 + 1.88801e7i 0.392896 + 0.680516i
950950 0 0
951951 −1.59024e7 −0.570180
952952 0 0
953953 −3.01356e7 −1.07485 −0.537424 0.843312i 0.680602π-0.680602\pi
−0.537424 + 0.843312i 0.680602π0.680602\pi
954954 0 0
955955 −1.27914e7 2.21554e7i −0.453848 0.786089i
956956 0 0
957957 3.79607e6 6.57499e6i 0.133985 0.232068i
958958 0 0
959959 1.49431e6 4.72474e6i 0.0524680 0.165894i
960960 0 0
961961 6.66107e6 1.15373e7i 0.232667 0.402992i
962962 0 0
963963 −903534. 1.56497e6i −0.0313963 0.0543800i
964964 0 0
965965 −2.15841e7 −0.746132
966966 0 0
967967 −2.88021e6 −0.0990509 −0.0495255 0.998773i 0.515771π-0.515771\pi
−0.0495255 + 0.998773i 0.515771π0.515771\pi
968968 0 0
969969 −5.63943e6 9.76779e6i −0.192942 0.334185i
970970 0 0
971971 −1.37245e7 + 2.37715e7i −0.467142 + 0.809113i −0.999295 0.0375347i 0.988050π-0.988050\pi
0.532154 + 0.846648i 0.321383π0.321383\pi
972972 0 0
973973 −9.29799e6 1.01650e7i −0.314852 0.344213i
974974 0 0
975975 2.86288e6 4.95866e6i 0.0964478 0.167052i
976976 0 0
977977 2.94262e7 + 5.09676e7i 0.986274 + 1.70828i 0.636133 + 0.771579i 0.280533π0.280533\pi
0.350141 + 0.936697i 0.386134π0.386134\pi
978978 0 0
979979 9.04164e6 0.301502
980980 0 0
981981 −6.78938e6 −0.225246
982982 0 0
983983 9.30384e6 + 1.61147e7i 0.307099 + 0.531911i 0.977726 0.209883i 0.0673084π-0.0673084\pi
−0.670627 + 0.741794i 0.733975π0.733975\pi
984984 0 0
985985 1.47442e7 2.55376e7i 0.484205 0.838668i
986986 0 0
987987 5.36643e6 + 5.86687e6i 0.175345 + 0.191696i
988988 0 0
989989 1.20231e7 2.08246e7i 0.390865 0.676997i
990990 0 0
991991 1.23577e7 + 2.14041e7i 0.399718 + 0.692331i 0.993691 0.112154i 0.0357749π-0.0357749\pi
−0.593973 + 0.804485i 0.702442π0.702442\pi
992992 0 0
993993 27470.5 0.000884083
994994 0 0
995995 5.26172e6 0.168488
996996 0 0
997997 8.97906e6 + 1.55522e7i 0.286084 + 0.495511i 0.972871 0.231347i 0.0743131π-0.0743131\pi
−0.686788 + 0.726858i 0.740980π0.740980\pi
998998 0 0
999999 5.94147e6 1.02909e7i 0.188356 0.326243i
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 336.6.q.e.289.1 4
4.3 odd 2 21.6.e.b.16.1 yes 4
7.4 even 3 inner 336.6.q.e.193.1 4
12.11 even 2 63.6.e.c.37.2 4
28.3 even 6 147.6.e.l.67.1 4
28.11 odd 6 21.6.e.b.4.1 4
28.19 even 6 147.6.a.k.1.2 2
28.23 odd 6 147.6.a.i.1.2 2
28.27 even 2 147.6.e.l.79.1 4
84.11 even 6 63.6.e.c.46.2 4
84.23 even 6 441.6.a.t.1.1 2
84.47 odd 6 441.6.a.s.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.1 4 28.11 odd 6
21.6.e.b.16.1 yes 4 4.3 odd 2
63.6.e.c.37.2 4 12.11 even 2
63.6.e.c.46.2 4 84.11 even 6
147.6.a.i.1.2 2 28.23 odd 6
147.6.a.k.1.2 2 28.19 even 6
147.6.e.l.67.1 4 28.3 even 6
147.6.e.l.79.1 4 28.27 even 2
336.6.q.e.193.1 4 7.4 even 3 inner
336.6.q.e.289.1 4 1.1 even 1 trivial
441.6.a.s.1.1 2 84.47 odd 6
441.6.a.t.1.1 2 84.23 even 6