Properties

Label 1620.2.e.b.971.12
Level 16201620
Weight 22
Character 1620.971
Analytic conductor 12.93612.936
Analytic rank 00
Dimension 4848
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1620,2,Mod(971,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.971");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1620=22345 1620 = 2^{2} \cdot 3^{4} \cdot 5
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1620.e (of order 22, degree 11, minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 12.935765127412.9357651274
Analytic rank: 00
Dimension: 4848
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 971.12
Character χ\chi == 1620.971
Dual form 1620.2.e.b.971.11

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+(1.04983+0.947554i)q2+(0.2042821.98954i)q41.00000iq5+4.17013iq7+(1.67074+2.28224i)q8+(0.947554+1.04983i)q10+6.44191q11+1.77467q13+(3.951424.37792i)q14+(3.916540.812853i)q16+0.950682iq17+2.19286iq19+(1.989540.204282i)q20+(6.76290+6.10406i)q22+2.02013q231.00000q25+(1.86310+1.68160i)q26+(8.29663+0.851880i)q286.52706iq29+2.84246iq31+(4.881922.85778i)q32+(0.9008230.998053i)q34+4.17013q355.83599q37+(2.077862.30213i)q38+(2.282241.67074i)q407.11552iq41+5.30982iq43+(1.3159612.8164i)q44+(2.12079+1.91418i)q460.403992q4710.3900q49+(1.049830.947554i)q50+(0.3625333.53078i)q52+4.87844iq536.44191iq55+(9.51725+6.96718i)q56+(6.18474+6.85230i)q58+2.68783q59+4.09304q61+(2.693382.98409i)q62+(2.41728+7.62606i)q641.77467iq65+10.8959iq67+(1.89142+0.194207i)q68+(4.37792+3.95142i)q702.23126q711.74662q73+(6.126795.52992i)q74+(4.36279+0.447962i)q76+26.8636iq770.247600iq79+(0.812853+3.91654i)q80+(6.74234+7.47008i)q82+11.6728q83+0.950682q85+(5.031355.57441i)q86+(10.7627+14.7020i)q88+6.13881iq89+7.40061iq91+(0.4126754.01913i)q92+(0.4241230.382804i)q94+2.19286q95+18.1701q97+(10.90779.84505i)q98+O(q100)q+(-1.04983 + 0.947554i) q^{2} +(0.204282 - 1.98954i) q^{4} -1.00000i q^{5} +4.17013i q^{7} +(1.67074 + 2.28224i) q^{8} +(0.947554 + 1.04983i) q^{10} +6.44191 q^{11} +1.77467 q^{13} +(-3.95142 - 4.37792i) q^{14} +(-3.91654 - 0.812853i) q^{16} +0.950682i q^{17} +2.19286i q^{19} +(-1.98954 - 0.204282i) q^{20} +(-6.76290 + 6.10406i) q^{22} +2.02013 q^{23} -1.00000 q^{25} +(-1.86310 + 1.68160i) q^{26} +(8.29663 + 0.851880i) q^{28} -6.52706i q^{29} +2.84246i q^{31} +(4.88192 - 2.85778i) q^{32} +(-0.900823 - 0.998053i) q^{34} +4.17013 q^{35} -5.83599 q^{37} +(-2.07786 - 2.30213i) q^{38} +(2.28224 - 1.67074i) q^{40} -7.11552i q^{41} +5.30982i q^{43} +(1.31596 - 12.8164i) q^{44} +(-2.12079 + 1.91418i) q^{46} -0.403992 q^{47} -10.3900 q^{49} +(1.04983 - 0.947554i) q^{50} +(0.362533 - 3.53078i) q^{52} +4.87844i q^{53} -6.44191i q^{55} +(-9.51725 + 6.96718i) q^{56} +(6.18474 + 6.85230i) q^{58} +2.68783 q^{59} +4.09304 q^{61} +(-2.69338 - 2.98409i) q^{62} +(-2.41728 + 7.62606i) q^{64} -1.77467i q^{65} +10.8959i q^{67} +(1.89142 + 0.194207i) q^{68} +(-4.37792 + 3.95142i) q^{70} -2.23126 q^{71} -1.74662 q^{73} +(6.12679 - 5.52992i) q^{74} +(4.36279 + 0.447962i) q^{76} +26.8636i q^{77} -0.247600i q^{79} +(-0.812853 + 3.91654i) q^{80} +(6.74234 + 7.47008i) q^{82} +11.6728 q^{83} +0.950682 q^{85} +(-5.03135 - 5.57441i) q^{86} +(10.7627 + 14.7020i) q^{88} +6.13881i q^{89} +7.40061i q^{91} +(0.412675 - 4.01913i) q^{92} +(0.424123 - 0.382804i) q^{94} +2.19286 q^{95} +18.1701 q^{97} +(10.9077 - 9.84505i) q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 48q48q25+12q34+12q4012q4648q49+36q52+36q5848q6424q7312q7636q8236q94+24q97+O(q100) 48 q - 48 q^{25} + 12 q^{34} + 12 q^{40} - 12 q^{46} - 48 q^{49} + 36 q^{52} + 36 q^{58} - 48 q^{64} - 24 q^{73} - 12 q^{76} - 36 q^{82} - 36 q^{94} + 24 q^{97}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1620Z)×\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times.

nn 811811 12971297 15411541
χ(n)\chi(n) 1-1 11 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 −1.04983 + 0.947554i −0.742341 + 0.670022i
33 0 0
44 0.204282 1.98954i 0.102141 0.994770i
55 1.00000i 0.447214i
66 0 0
77 4.17013i 1.57616i 0.615573 + 0.788080i 0.288925π0.288925\pi
−0.615573 + 0.788080i 0.711075π0.711075\pi
88 1.67074 + 2.28224i 0.590694 + 0.806895i
99 0 0
1010 0.947554 + 1.04983i 0.299643 + 0.331985i
1111 6.44191 1.94231 0.971154 0.238453i 0.0766403π-0.0766403\pi
0.971154 + 0.238453i 0.0766403π0.0766403\pi
1212 0 0
1313 1.77467 0.492205 0.246103 0.969244i 0.420850π-0.420850\pi
0.246103 + 0.969244i 0.420850π0.420850\pi
1414 −3.95142 4.37792i −1.05606 1.17005i
1515 0 0
1616 −3.91654 0.812853i −0.979135 0.203213i
1717 0.950682i 0.230574i 0.993332 + 0.115287i 0.0367788π0.0367788\pi
−0.993332 + 0.115287i 0.963221π0.963221\pi
1818 0 0
1919 2.19286i 0.503078i 0.967847 + 0.251539i 0.0809366π0.0809366\pi
−0.967847 + 0.251539i 0.919063π0.919063\pi
2020 −1.98954 0.204282i −0.444875 0.0456788i
2121 0 0
2222 −6.76290 + 6.10406i −1.44186 + 1.30139i
2323 2.02013 0.421226 0.210613 0.977569i 0.432454π-0.432454\pi
0.210613 + 0.977569i 0.432454π0.432454\pi
2424 0 0
2525 −1.00000 −0.200000
2626 −1.86310 + 1.68160i −0.365384 + 0.329788i
2727 0 0
2828 8.29663 + 0.851880i 1.56792 + 0.160990i
2929 6.52706i 1.21204i −0.795448 0.606022i 0.792764π-0.792764\pi
0.795448 0.606022i 0.207236π-0.207236\pi
3030 0 0
3131 2.84246i 0.510521i 0.966872 + 0.255260i 0.0821612π0.0821612\pi
−0.966872 + 0.255260i 0.917839π0.917839\pi
3232 4.88192 2.85778i 0.863009 0.505188i
3333 0 0
3434 −0.900823 0.998053i −0.154490 0.171165i
3535 4.17013 0.704880
3636 0 0
3737 −5.83599 −0.959431 −0.479715 0.877424i 0.659260π-0.659260\pi
−0.479715 + 0.877424i 0.659260π0.659260\pi
3838 −2.07786 2.30213i −0.337073 0.373455i
3939 0 0
4040 2.28224 1.67074i 0.360855 0.264167i
4141 7.11552i 1.11126i −0.831430 0.555629i 0.812478π-0.812478\pi
0.831430 0.555629i 0.187522π-0.187522\pi
4242 0 0
4343 5.30982i 0.809741i 0.914374 + 0.404870i 0.132683π0.132683\pi
−0.914374 + 0.404870i 0.867317π0.867317\pi
4444 1.31596 12.8164i 0.198389 1.93215i
4545 0 0
4646 −2.12079 + 1.91418i −0.312694 + 0.282231i
4747 −0.403992 −0.0589283 −0.0294641 0.999566i 0.509380π-0.509380\pi
−0.0294641 + 0.999566i 0.509380π0.509380\pi
4848 0 0
4949 −10.3900 −1.48428
5050 1.04983 0.947554i 0.148468 0.134004i
5151 0 0
5252 0.362533 3.53078i 0.0502743 0.489631i
5353 4.87844i 0.670106i 0.942199 + 0.335053i 0.108754π0.108754\pi
−0.942199 + 0.335053i 0.891246π0.891246\pi
5454 0 0
5555 6.44191i 0.868627i
5656 −9.51725 + 6.96718i −1.27180 + 0.931029i
5757 0 0
5858 6.18474 + 6.85230i 0.812097 + 0.899751i
5959 2.68783 0.349926 0.174963 0.984575i 0.444019π-0.444019\pi
0.174963 + 0.984575i 0.444019π0.444019\pi
6060 0 0
6161 4.09304 0.524060 0.262030 0.965060i 0.415608π-0.415608\pi
0.262030 + 0.965060i 0.415608π0.415608\pi
6262 −2.69338 2.98409i −0.342060 0.378980i
6363 0 0
6464 −2.41728 + 7.62606i −0.302160 + 0.953257i
6565 1.77467i 0.220121i
6666 0 0
6767 10.8959i 1.33115i 0.746331 + 0.665575i 0.231814π0.231814\pi
−0.746331 + 0.665575i 0.768186π0.768186\pi
6868 1.89142 + 0.194207i 0.229368 + 0.0235510i
6969 0 0
7070 −4.37792 + 3.95142i −0.523262 + 0.472285i
7171 −2.23126 −0.264802 −0.132401 0.991196i 0.542269π-0.542269\pi
−0.132401 + 0.991196i 0.542269π0.542269\pi
7272 0 0
7373 −1.74662 −0.204426 −0.102213 0.994763i 0.532592π-0.532592\pi
−0.102213 + 0.994763i 0.532592π0.532592\pi
7474 6.12679 5.52992i 0.712225 0.642840i
7575 0 0
7676 4.36279 + 0.447962i 0.500447 + 0.0513848i
7777 26.8636i 3.06139i
7878 0 0
7979 0.247600i 0.0278572i −0.999903 0.0139286i 0.995566π-0.995566\pi
0.999903 0.0139286i 0.00443375π-0.00443375\pi
8080 −0.812853 + 3.91654i −0.0908797 + 0.437882i
8181 0 0
8282 6.74234 + 7.47008i 0.744567 + 0.824932i
8383 11.6728 1.28126 0.640629 0.767851i 0.278674π-0.278674\pi
0.640629 + 0.767851i 0.278674π0.278674\pi
8484 0 0
8585 0.950682 0.103116
8686 −5.03135 5.57441i −0.542544 0.601104i
8787 0 0
8888 10.7627 + 14.7020i 1.14731 + 1.56724i
8989 6.13881i 0.650713i 0.945591 + 0.325356i 0.105484π0.105484\pi
−0.945591 + 0.325356i 0.894516π0.894516\pi
9090 0 0
9191 7.40061i 0.775794i
9292 0.412675 4.01913i 0.0430244 0.419023i
9393 0 0
9494 0.424123 0.382804i 0.0437449 0.0394833i
9595 2.19286 0.224983
9696 0 0
9797 18.1701 1.84489 0.922446 0.386127i 0.126187π-0.126187\pi
0.922446 + 0.386127i 0.126187π0.126187\pi
9898 10.9077 9.84505i 1.10184 0.994500i
9999 0 0
100100 −0.204282 + 1.98954i −0.0204282 + 0.198954i
101101 10.2399i 1.01891i 0.860499 + 0.509453i 0.170152π0.170152\pi
−0.860499 + 0.509453i 0.829848π0.829848\pi
102102 0 0
103103 9.65985i 0.951814i −0.879496 0.475907i 0.842120π-0.842120\pi
0.879496 0.475907i 0.157880π-0.157880\pi
104104 2.96501 + 4.05023i 0.290743 + 0.397158i
105105 0 0
106106 −4.62259 5.12153i −0.448986 0.497447i
107107 3.23356 0.312600 0.156300 0.987710i 0.450043π-0.450043\pi
0.156300 + 0.987710i 0.450043π0.450043\pi
108108 0 0
109109 2.88351 0.276190 0.138095 0.990419i 0.455902π-0.455902\pi
0.138095 + 0.990419i 0.455902π0.455902\pi
110110 6.10406 + 6.76290i 0.581999 + 0.644817i
111111 0 0
112112 3.38970 16.3325i 0.320297 1.54327i
113113 3.08823i 0.290516i 0.989394 + 0.145258i 0.0464012π0.0464012\pi
−0.989394 + 0.145258i 0.953599π0.953599\pi
114114 0 0
115115 2.02013i 0.188378i
116116 −12.9858 1.33336i −1.20571 0.123799i
117117 0 0
118118 −2.82176 + 2.54686i −0.259764 + 0.234458i
119119 −3.96446 −0.363422
120120 0 0
121121 30.4982 2.77256
122122 −4.29699 + 3.87838i −0.389031 + 0.351132i
123123 0 0
124124 5.65518 + 0.580662i 0.507850 + 0.0521450i
125125 1.00000i 0.0894427i
126126 0 0
127127 2.16321i 0.191954i −0.995384 0.0959771i 0.969402π-0.969402\pi
0.995384 0.0959771i 0.0305976π-0.0305976\pi
128128 −4.68837 10.2966i −0.414398 0.910096i
129129 0 0
130130 1.68160 + 1.86310i 0.147486 + 0.163405i
131131 1.91441 0.167263 0.0836315 0.996497i 0.473348π-0.473348\pi
0.0836315 + 0.996497i 0.473348π0.473348\pi
132132 0 0
133133 −9.14452 −0.792931
134134 −10.3245 11.4389i −0.891901 0.988168i
135135 0 0
136136 −2.16969 + 1.58834i −0.186049 + 0.136199i
137137 11.6608i 0.996250i 0.867105 + 0.498125i 0.165978π0.165978\pi
−0.867105 + 0.498125i 0.834022π0.834022\pi
138138 0 0
139139 7.66107i 0.649804i 0.945748 + 0.324902i 0.105331π0.105331\pi
−0.945748 + 0.324902i 0.894669π0.894669\pi
140140 0.851880 8.29663i 0.0719970 0.701194i
141141 0 0
142142 2.34244 2.11424i 0.196574 0.177423i
143143 11.4323 0.956014
144144 0 0
145145 −6.52706 −0.542043
146146 1.83365 1.65501i 0.151754 0.136970i
147147 0 0
148148 −1.19219 + 11.6109i −0.0979970 + 0.954413i
149149 3.49438i 0.286271i −0.989703 0.143135i 0.954282π-0.954282\pi
0.989703 0.143135i 0.0457184π-0.0457184\pi
150150 0 0
151151 7.11211i 0.578775i 0.957212 + 0.289388i 0.0934517π0.0934517\pi
−0.957212 + 0.289388i 0.906548π0.906548\pi
152152 −5.00465 + 3.66370i −0.405931 + 0.297165i
153153 0 0
154154 −25.4547 28.2022i −2.05120 2.27259i
155155 2.84246 0.228312
156156 0 0
157157 −3.15875 −0.252096 −0.126048 0.992024i 0.540229π-0.540229\pi
−0.126048 + 0.992024i 0.540229π0.540229\pi
158158 0.234614 + 0.259938i 0.0186649 + 0.0206795i
159159 0 0
160160 −2.85778 4.88192i −0.225927 0.385949i
161161 8.42420i 0.663920i
162162 0 0
163163 14.1062i 1.10488i 0.833551 + 0.552442i 0.186304π0.186304\pi
−0.833551 + 0.552442i 0.813696π0.813696\pi
164164 −14.1566 1.45357i −1.10545 0.113505i
165165 0 0
166166 −12.2545 + 11.0606i −0.951131 + 0.858471i
167167 −18.3328 −1.41863 −0.709317 0.704890i 0.750997π-0.750997\pi
−0.709317 + 0.704890i 0.750997π0.750997\pi
168168 0 0
169169 −9.85054 −0.757734
170170 −0.998053 + 0.900823i −0.0765472 + 0.0690899i
171171 0 0
172172 10.5641 + 1.08470i 0.805506 + 0.0827076i
173173 7.88176i 0.599239i 0.954059 + 0.299620i 0.0968598π0.0968598\pi
−0.954059 + 0.299620i 0.903140π0.903140\pi
174174 0 0
175175 4.17013i 0.315232i
176176 −25.2300 5.23632i −1.90178 0.394703i
177177 0 0
178178 −5.81686 6.44470i −0.435992 0.483051i
179179 −9.84108 −0.735557 −0.367778 0.929914i 0.619881π-0.619881\pi
−0.367778 + 0.929914i 0.619881π0.619881\pi
180180 0 0
181181 18.3215 1.36183 0.680913 0.732364i 0.261583π-0.261583\pi
0.680913 + 0.732364i 0.261583π0.261583\pi
182182 −7.01248 7.76937i −0.519799 0.575904i
183183 0 0
184184 3.37510 + 4.61043i 0.248816 + 0.339885i
185185 5.83599i 0.429070i
186186 0 0
187187 6.12420i 0.447846i
188188 −0.0825282 + 0.803758i −0.00601898 + 0.0586201i
189189 0 0
190190 −2.30213 + 2.07786i −0.167014 + 0.150744i
191191 −19.0805 −1.38061 −0.690307 0.723516i 0.742525π-0.742525\pi
−0.690307 + 0.723516i 0.742525π0.742525\pi
192192 0 0
193193 −13.0031 −0.935983 −0.467991 0.883733i 0.655022π-0.655022\pi
−0.467991 + 0.883733i 0.655022π0.655022\pi
194194 −19.0755 + 17.2171i −1.36954 + 1.23612i
195195 0 0
196196 −2.12248 + 20.6712i −0.151606 + 1.47652i
197197 15.8462i 1.12899i −0.825436 0.564496i 0.809071π-0.809071\pi
0.825436 0.564496i 0.190929π-0.190929\pi
198198 0 0
199199 1.24563i 0.0883001i 0.999025 + 0.0441500i 0.0140580π0.0140580\pi
−0.999025 + 0.0441500i 0.985942π0.985942\pi
200200 −1.67074 2.28224i −0.118139 0.161379i
201201 0 0
202202 −9.70284 10.7501i −0.682689 0.756375i
203203 27.2187 1.91038
204204 0 0
205205 −7.11552 −0.496970
206206 9.15324 + 10.1412i 0.637736 + 0.706571i
207207 0 0
208208 −6.95057 1.44255i −0.481935 0.100023i
209209 14.1262i 0.977132i
210210 0 0
211211 16.3244i 1.12382i −0.827198 0.561910i 0.810067π-0.810067\pi
0.827198 0.561910i 0.189933π-0.189933\pi
212212 9.70586 + 0.996577i 0.666601 + 0.0684452i
213213 0 0
214214 −3.39469 + 3.06398i −0.232056 + 0.209449i
215215 5.30982 0.362127
216216 0 0
217217 −11.8534 −0.804662
218218 −3.02719 + 2.73228i −0.205027 + 0.185053i
219219 0 0
220220 −12.8164 1.31596i −0.864084 0.0887222i
221221 1.68715i 0.113490i
222222 0 0
223223 17.9257i 1.20040i −0.799851 0.600198i 0.795088π-0.795088\pi
0.799851 0.600198i 0.204912π-0.204912\pi
224224 11.9173 + 20.3582i 0.796257 + 1.36024i
225225 0 0
226226 −2.92626 3.24211i −0.194652 0.215662i
227227 10.4304 0.692293 0.346146 0.938180i 0.387490π-0.387490\pi
0.346146 + 0.938180i 0.387490π0.387490\pi
228228 0 0
229229 −18.7878 −1.24153 −0.620765 0.783997i 0.713178π-0.713178\pi
−0.620765 + 0.783997i 0.713178π0.713178\pi
230230 1.91418 + 2.12079i 0.126217 + 0.139841i
231231 0 0
232232 14.8963 10.9050i 0.977993 0.715948i
233233 7.28693i 0.477382i −0.971096 0.238691i 0.923282π-0.923282\pi
0.971096 0.238691i 0.0767184π-0.0767184\pi
234234 0 0
235235 0.403992i 0.0263535i
236236 0.549074 5.34754i 0.0357417 0.348096i
237237 0 0
238238 4.16201 3.75654i 0.269783 0.243501i
239239 29.6817 1.91995 0.959973 0.280091i 0.0903647π-0.0903647\pi
0.959973 + 0.280091i 0.0903647π0.0903647\pi
240240 0 0
241241 −12.7503 −0.821321 −0.410661 0.911788i 0.634702π-0.634702\pi
−0.410661 + 0.911788i 0.634702π0.634702\pi
242242 −32.0179 + 28.8987i −2.05819 + 1.85768i
243243 0 0
244244 0.836133 8.14327i 0.0535279 0.521319i
245245 10.3900i 0.663790i
246246 0 0
247247 3.89161i 0.247618i
248248 −6.48718 + 4.74900i −0.411937 + 0.301562i
249249 0 0
250250 −0.947554 1.04983i −0.0599286 0.0663970i
251251 −2.89830 −0.182939 −0.0914694 0.995808i 0.529156π-0.529156\pi
−0.0914694 + 0.995808i 0.529156π0.529156\pi
252252 0 0
253253 13.0135 0.818151
254254 2.04976 + 2.27101i 0.128614 + 0.142496i
255255 0 0
256256 14.6785 + 6.36714i 0.917409 + 0.397946i
257257 9.96613i 0.621670i −0.950464 0.310835i 0.899391π-0.899391\pi
0.950464 0.310835i 0.100609π-0.100609\pi
258258 0 0
259259 24.3368i 1.51222i
260260 −3.53078 0.362533i −0.218970 0.0224833i
261261 0 0
262262 −2.00981 + 1.81401i −0.124166 + 0.112070i
263263 1.22162 0.0753281 0.0376640 0.999290i 0.488008π-0.488008\pi
0.0376640 + 0.999290i 0.488008π0.488008\pi
264264 0 0
265265 4.87844 0.299680
266266 9.60019 8.66493i 0.588625 0.531281i
267267 0 0
268268 21.6779 + 2.22584i 1.32419 + 0.135965i
269269 19.0910i 1.16400i 0.813190 + 0.581998i 0.197729π0.197729\pi
−0.813190 + 0.581998i 0.802271π0.802271\pi
270270 0 0
271271 12.0019i 0.729061i −0.931191 0.364531i 0.881229π-0.881229\pi
0.931191 0.364531i 0.118771π-0.118771\pi
272272 0.772764 3.72338i 0.0468557 0.225763i
273273 0 0
274274 −11.0492 12.2418i −0.667509 0.739557i
275275 −6.44191 −0.388462
276276 0 0
277277 −24.0283 −1.44372 −0.721860 0.692039i 0.756712π-0.756712\pi
−0.721860 + 0.692039i 0.756712π0.756712\pi
278278 −7.25928 8.04282i −0.435383 0.482376i
279279 0 0
280280 6.96718 + 9.51725i 0.416369 + 0.568764i
281281 1.54857i 0.0923797i −0.998933 0.0461898i 0.985292π-0.985292\pi
0.998933 0.0461898i 0.0147079π-0.0147079\pi
282282 0 0
283283 24.5575i 1.45979i 0.683559 + 0.729895i 0.260431π0.260431\pi
−0.683559 + 0.729895i 0.739569π0.739569\pi
284284 −0.455806 + 4.43919i −0.0270471 + 0.263417i
285285 0 0
286286 −12.0019 + 10.8327i −0.709689 + 0.640551i
287287 29.6726 1.75152
288288 0 0
289289 16.0962 0.946836
290290 6.85230 6.18474i 0.402381 0.363181i
291291 0 0
292292 −0.356802 + 3.47496i −0.0208802 + 0.203357i
293293 27.1687i 1.58721i −0.608431 0.793607i 0.708201π-0.708201\pi
0.608431 0.793607i 0.291799π-0.291799\pi
294294 0 0
295295 2.68783i 0.156492i
296296 −9.75040 13.3192i −0.566730 0.774160i
297297 0 0
298298 3.31111 + 3.66850i 0.191808 + 0.212511i
299299 3.58507 0.207330
300300 0 0
301301 −22.1426 −1.27628
302302 −6.73911 7.46650i −0.387792 0.429649i
303303 0 0
304304 1.78248 8.58844i 0.102232 0.492581i
305305 4.09304i 0.234367i
306306 0 0
307307 3.96646i 0.226378i −0.993573 0.113189i 0.963893π-0.963893\pi
0.993573 0.113189i 0.0361065π-0.0361065\pi
308308 53.4461 + 5.48773i 3.04538 + 0.312693i
309309 0 0
310310 −2.98409 + 2.69338i −0.169485 + 0.152974i
311311 −19.3841 −1.09917 −0.549586 0.835437i 0.685215π-0.685215\pi
−0.549586 + 0.835437i 0.685215π0.685215\pi
312312 0 0
313313 21.9702 1.24183 0.620915 0.783878i 0.286761π-0.286761\pi
0.620915 + 0.783878i 0.286761π0.286761\pi
314314 3.31615 2.99309i 0.187141 0.168910i
315315 0 0
316316 −0.492610 0.0505801i −0.0277115 0.00284535i
317317 26.6227i 1.49528i −0.664104 0.747640i 0.731187π-0.731187\pi
0.664104 0.747640i 0.268813π-0.268813\pi
318318 0 0
319319 42.0467i 2.35416i
320320 7.62606 + 2.41728i 0.426310 + 0.135130i
321321 0 0
322322 −7.98238 8.84397i −0.444841 0.492855i
323323 −2.08472 −0.115997
324324 0 0
325325 −1.77467 −0.0984411
326326 −13.3664 14.8091i −0.740297 0.820201i
327327 0 0
328328 16.2394 11.8882i 0.896669 0.656414i
329329 1.68470i 0.0928804i
330330 0 0
331331 33.2166i 1.82575i −0.408238 0.912876i 0.633857π-0.633857\pi
0.408238 0.912876i 0.366143π-0.366143\pi
332332 2.38454 23.2235i 0.130869 1.27456i
333333 0 0
334334 19.2463 17.3713i 1.05311 0.950516i
335335 10.8959 0.595309
336336 0 0
337337 1.61717 0.0880927 0.0440463 0.999029i 0.485975π-0.485975\pi
0.0440463 + 0.999029i 0.485975π0.485975\pi
338338 10.3414 9.33392i 0.562497 0.507698i
339339 0 0
340340 0.194207 1.89142i 0.0105323 0.102577i
341341 18.3109i 0.991588i
342342 0 0
343343 14.1366i 0.763302i
344344 −12.1183 + 8.87132i −0.653376 + 0.478309i
345345 0 0
346346 −7.46840 8.27450i −0.401503 0.444840i
347347 5.18596 0.278397 0.139198 0.990265i 0.455547π-0.455547\pi
0.139198 + 0.990265i 0.455547π0.455547\pi
348348 0 0
349349 −18.3343 −0.981414 −0.490707 0.871325i 0.663261π-0.663261\pi
−0.490707 + 0.871325i 0.663261π0.663261\pi
350350 3.95142 + 4.37792i 0.211212 + 0.234010i
351351 0 0
352352 31.4489 18.4095i 1.67623 0.981231i
353353 10.6975i 0.569368i −0.958621 0.284684i 0.908111π-0.908111\pi
0.958621 0.284684i 0.0918886π-0.0918886\pi
354354 0 0
355355 2.23126i 0.118423i
356356 12.2134 + 1.25405i 0.647309 + 0.0664643i
357357 0 0
358358 10.3314 9.32495i 0.546034 0.492839i
359359 15.1958 0.802005 0.401002 0.916077i 0.368662π-0.368662\pi
0.401002 + 0.916077i 0.368662π0.368662\pi
360360 0 0
361361 14.1913 0.746913
362362 −19.2344 + 17.3606i −1.01094 + 0.912454i
363363 0 0
364364 14.7238 + 1.51181i 0.771737 + 0.0792403i
365365 1.74662i 0.0914221i
366366 0 0
367367 24.5582i 1.28193i 0.767571 + 0.640964i 0.221465π0.221465\pi
−0.767571 + 0.640964i 0.778535π0.778535\pi
368368 −7.91192 1.64207i −0.412437 0.0855987i
369369 0 0
370370 −5.52992 6.12679i −0.287487 0.318517i
371371 −20.3437 −1.05619
372372 0 0
373373 18.6090 0.963538 0.481769 0.876298i 0.339994π-0.339994\pi
0.481769 + 0.876298i 0.339994π0.339994\pi
374374 −5.80302 6.42937i −0.300067 0.332455i
375375 0 0
376376 −0.674964 0.922009i −0.0348086 0.0475490i
377377 11.5834i 0.596575i
378378 0 0
379379 27.9052i 1.43339i 0.697386 + 0.716696i 0.254347π0.254347\pi
−0.697386 + 0.716696i 0.745653π0.745653\pi
380380 0.447962 4.36279i 0.0229800 0.223806i
381381 0 0
382382 20.0312 18.0798i 1.02489 0.925042i
383383 7.40233 0.378241 0.189121 0.981954i 0.439436π-0.439436\pi
0.189121 + 0.981954i 0.439436π0.439436\pi
384384 0 0
385385 26.8636 1.36909
386386 13.6510 12.3211i 0.694818 0.627129i
387387 0 0
388388 3.71181 36.1501i 0.188439 1.83524i
389389 17.9397i 0.909579i 0.890599 + 0.454789i 0.150285π0.150285\pi
−0.890599 + 0.454789i 0.849715π0.849715\pi
390390 0 0
391391 1.92050i 0.0971239i
392392 −17.3589 23.7124i −0.876756 1.19766i
393393 0 0
394394 15.0151 + 16.6358i 0.756450 + 0.838097i
395395 −0.247600 −0.0124581
396396 0 0
397397 −27.9101 −1.40077 −0.700383 0.713768i 0.746987π-0.746987\pi
−0.700383 + 0.713768i 0.746987π0.746987\pi
398398 −1.18030 1.30769i −0.0591630 0.0655488i
399399 0 0
400400 3.91654 + 0.812853i 0.195827 + 0.0406426i
401401 3.99705i 0.199603i 0.995007 + 0.0998015i 0.0318208π0.0318208\pi
−0.995007 + 0.0998015i 0.968179π0.968179\pi
402402 0 0
403403 5.04443i 0.251281i
404404 20.3726 + 2.09182i 1.01358 + 0.104072i
405405 0 0
406406 −28.5749 + 25.7912i −1.41815 + 1.27999i
407407 −37.5949 −1.86351
408408 0 0
409409 22.1291 1.09421 0.547107 0.837063i 0.315729π-0.315729\pi
0.547107 + 0.837063i 0.315729π0.315729\pi
410410 7.47008 6.74234i 0.368921 0.332981i
411411 0 0
412412 −19.2187 1.97333i −0.946836 0.0972190i
413413 11.2086i 0.551539i
414414 0 0
415415 11.6728i 0.572996i
416416 8.66380 5.07161i 0.424778 0.248656i
417417 0 0
418418 −13.3854 14.8301i −0.654700 0.725365i
419419 −31.6137 −1.54443 −0.772215 0.635362i 0.780851π-0.780851\pi
−0.772215 + 0.635362i 0.780851π0.780851\pi
420420 0 0
421421 30.0566 1.46487 0.732435 0.680837i 0.238384π-0.238384\pi
0.732435 + 0.680837i 0.238384π0.238384\pi
422422 15.4683 + 17.1379i 0.752984 + 0.834258i
423423 0 0
424424 −11.1338 + 8.15059i −0.540705 + 0.395828i
425425 0.950682i 0.0461148i
426426 0 0
427427 17.0685i 0.826002i
428428 0.660558 6.43331i 0.0319293 0.310966i
429429 0 0
430430 −5.57441 + 5.03135i −0.268822 + 0.242633i
431431 −25.0832 −1.20822 −0.604108 0.796902i 0.706471π-0.706471\pi
−0.604108 + 0.796902i 0.706471π0.706471\pi
432432 0 0
433433 29.4297 1.41430 0.707151 0.707062i 0.249980π-0.249980\pi
0.707151 + 0.707062i 0.249980π0.249980\pi
434434 12.4441 11.2318i 0.597334 0.539141i
435435 0 0
436436 0.589048 5.73685i 0.0282103 0.274745i
437437 4.42987i 0.211909i
438438 0 0
439439 28.6552i 1.36764i 0.729651 + 0.683819i 0.239682π0.239682\pi
−0.729651 + 0.683819i 0.760318π0.760318\pi
440440 14.7020 10.7627i 0.700891 0.513093i
441441 0 0
442442 −1.59866 1.77122i −0.0760407 0.0842482i
443443 29.8579 1.41859 0.709297 0.704910i 0.249013π-0.249013\pi
0.709297 + 0.704910i 0.249013π0.249013\pi
444444 0 0
445445 6.13881 0.291008
446446 16.9856 + 18.8190i 0.804292 + 0.891104i
447447 0 0
448448 −31.8016 10.0804i −1.50249 0.476252i
449449 14.7017i 0.693815i −0.937899 0.346908i 0.887232π-0.887232\pi
0.937899 0.346908i 0.112768π-0.112768\pi
450450 0 0
451451 45.8375i 2.15841i
452452 6.14415 + 0.630868i 0.288997 + 0.0296735i
453453 0 0
454454 −10.9502 + 9.88341i −0.513918 + 0.463852i
455455 7.40061 0.346946
456456 0 0
457457 −8.83336 −0.413207 −0.206604 0.978425i 0.566241π-0.566241\pi
−0.206604 + 0.978425i 0.566241π0.566241\pi
458458 19.7239 17.8024i 0.921639 0.831853i
459459 0 0
460460 −4.01913 0.412675i −0.187393 0.0192411i
461461 0.913372i 0.0425400i −0.999774 0.0212700i 0.993229π-0.993229\pi
0.999774 0.0212700i 0.00677096π-0.00677096\pi
462462 0 0
463463 7.87134i 0.365812i −0.983130 0.182906i 0.941450π-0.941450\pi
0.983130 0.182906i 0.0585505π-0.0585505\pi
464464 −5.30554 + 25.5635i −0.246303 + 1.18675i
465465 0 0
466466 6.90476 + 7.65003i 0.319857 + 0.354381i
467467 −28.6765 −1.32699 −0.663495 0.748181i 0.730928π-0.730928\pi
−0.663495 + 0.748181i 0.730928π0.730928\pi
468468 0 0
469469 −45.4375 −2.09811
470470 −0.382804 0.424123i −0.0176575 0.0195633i
471471 0 0
472472 4.49066 + 6.13429i 0.206699 + 0.282353i
473473 34.2054i 1.57277i
474474 0 0
475475 2.19286i 0.100616i
476476 −0.809867 + 7.88746i −0.0371202 + 0.361521i
477477 0 0
478478 −31.1607 + 28.1250i −1.42526 + 1.28641i
479479 26.3817 1.20541 0.602706 0.797963i 0.294089π-0.294089\pi
0.602706 + 0.797963i 0.294089π0.294089\pi
480480 0 0
481481 −10.3570 −0.472237
482482 13.3857 12.0816i 0.609701 0.550304i
483483 0 0
484484 6.23022 60.6773i 0.283192 2.75806i
485485 18.1701i 0.825060i
486486 0 0
487487 7.87222i 0.356724i −0.983965 0.178362i 0.942920π-0.942920\pi
0.983965 0.178362i 0.0570799π-0.0570799\pi
488488 6.83839 + 9.34132i 0.309559 + 0.422862i
489489 0 0
490490 −9.84505 10.9077i −0.444754 0.492759i
491491 −19.4940 −0.879751 −0.439876 0.898059i 0.644977π-0.644977\pi
−0.439876 + 0.898059i 0.644977π0.644977\pi
492492 0 0
493493 6.20516 0.279466
494494 −3.68752 4.08553i −0.165909 0.183817i
495495 0 0
496496 2.31050 11.1326i 0.103745 0.499868i
497497 9.30465i 0.417371i
498498 0 0
499499 0.846463i 0.0378929i −0.999821 0.0189464i 0.993969π-0.993969\pi
0.999821 0.0189464i 0.00603120π-0.00603120\pi
500500 1.98954 + 0.204282i 0.0889749 + 0.00913575i
501501 0 0
502502 3.04272 2.74629i 0.135803 0.122573i
503503 18.5253 0.826002 0.413001 0.910731i 0.364481π-0.364481\pi
0.413001 + 0.910731i 0.364481π0.364481\pi
504504 0 0
505505 10.2399 0.455668
506506 −13.6619 + 12.3310i −0.607347 + 0.548179i
507507 0 0
508508 −4.30380 0.441905i −0.190950 0.0196064i
509509 27.3040i 1.21023i −0.796138 0.605115i 0.793127π-0.793127\pi
0.796138 0.605115i 0.206873π-0.206873\pi
510510 0 0
511511 7.28361i 0.322208i
512512 −21.4432 + 7.22431i −0.947663 + 0.319272i
513513 0 0
514514 9.44345 + 10.4627i 0.416533 + 0.461491i
515515 −9.65985 −0.425664
516516 0 0
517517 −2.60248 −0.114457
518518 23.0605 + 25.5495i 1.01322 + 1.12258i
519519 0 0
520520 4.05023 2.96501i 0.177615 0.130024i
521521 25.9868i 1.13850i −0.822163 0.569252i 0.807233π-0.807233\pi
0.822163 0.569252i 0.192767π-0.192767\pi
522522 0 0
523523 30.1683i 1.31917i 0.751631 + 0.659584i 0.229268π0.229268\pi
−0.751631 + 0.659584i 0.770732π0.770732\pi
524524 0.391079 3.80880i 0.0170844 0.166388i
525525 0 0
526526 −1.28249 + 1.15755i −0.0559191 + 0.0504715i
527527 −2.70227 −0.117713
528528 0 0
529529 −18.9191 −0.822568
530530 −5.12153 + 4.62259i −0.222465 + 0.200793i
531531 0 0
532532 −1.86806 + 18.1934i −0.0809906 + 0.788784i
533533 12.6277i 0.546967i
534534 0 0
535535 3.23356i 0.139799i
536536 −24.8672 + 18.2042i −1.07410 + 0.786304i
537537 0 0
538538 −18.0897 20.0423i −0.779904 0.864083i
539539 −66.9311 −2.88293
540540 0 0
541541 −34.2068 −1.47067 −0.735333 0.677706i 0.762974π-0.762974\pi
−0.735333 + 0.677706i 0.762974π0.762974\pi
542542 11.3724 + 12.5999i 0.488487 + 0.541212i
543543 0 0
544544 2.71684 + 4.64115i 0.116483 + 0.198988i
545545 2.88351i 0.123516i
546546 0 0
547547 16.0152i 0.684761i −0.939561 0.342380i 0.888767π-0.888767\pi
0.939561 0.342380i 0.111233π-0.111233\pi
548548 23.1996 + 2.38209i 0.991039 + 0.101758i
549549 0 0
550550 6.76290 6.10406i 0.288371 0.260278i
551551 14.3130 0.609753
552552 0 0
553553 1.03252 0.0439073
554554 25.2256 22.7681i 1.07173 0.967324i
555555 0 0
556556 15.2420 + 1.56502i 0.646405 + 0.0663715i
557557 16.0344i 0.679401i −0.940534 0.339701i 0.889674π-0.889674\pi
0.940534 0.339701i 0.110326π-0.110326\pi
558558 0 0
559559 9.42320i 0.398559i
560560 −16.3325 3.38970i −0.690172 0.143241i
561561 0 0
562562 1.46735 + 1.62573i 0.0618964 + 0.0685772i
563563 4.47364 0.188541 0.0942707 0.995547i 0.469948π-0.469948\pi
0.0942707 + 0.995547i 0.469948π0.469948\pi
564564 0 0
565565 3.08823 0.129923
566566 −23.2695 25.7812i −0.978092 1.08366i
567567 0 0
568568 −3.72785 5.09229i −0.156417 0.213668i
569569 41.0451i 1.72070i −0.509705 0.860349i 0.670245π-0.670245\pi
0.509705 0.860349i 0.329755π-0.329755\pi
570570 0 0
571571 45.2501i 1.89366i −0.321741 0.946828i 0.604268π-0.604268\pi
0.321741 0.946828i 0.395732π-0.395732\pi
572572 2.33540 22.7450i 0.0976481 0.951014i
573573 0 0
574574 −31.1512 + 28.1164i −1.30023 + 1.17356i
575575 −2.02013 −0.0842452
576576 0 0
577577 −27.1400 −1.12985 −0.564927 0.825141i 0.691096π-0.691096\pi
−0.564927 + 0.825141i 0.691096π0.691096\pi
578578 −16.8983 + 15.2520i −0.702875 + 0.634401i
579579 0 0
580580 −1.33336 + 12.9858i −0.0553647 + 0.539208i
581581 48.6771i 2.01947i
582582 0 0
583583 31.4265i 1.30155i
584584 −2.91814 3.98621i −0.120753 0.164950i
585585 0 0
586586 25.7438 + 28.5225i 1.06347 + 1.17825i
587587 17.5383 0.723883 0.361941 0.932201i 0.382114π-0.382114\pi
0.361941 + 0.932201i 0.382114π0.382114\pi
588588 0 0
589589 −6.23313 −0.256831
590590 2.54686 + 2.82176i 0.104853 + 0.116170i
591591 0 0
592592 22.8569 + 4.74380i 0.939412 + 0.194969i
593593 15.6793i 0.643872i 0.946761 + 0.321936i 0.104334π0.104334\pi
−0.946761 + 0.321936i 0.895666π0.895666\pi
594594 0 0
595595 3.96446i 0.162527i
596596 −6.95221 0.713837i −0.284773 0.0292399i
597597 0 0
598598 −3.76371 + 3.39705i −0.153909 + 0.138916i
599599 −7.21949 −0.294980 −0.147490 0.989064i 0.547119π-0.547119\pi
−0.147490 + 0.989064i 0.547119π0.547119\pi
600600 0 0
601601 45.9916 1.87604 0.938019 0.346583i 0.112658π-0.112658\pi
0.938019 + 0.346583i 0.112658π0.112658\pi
602602 23.2460 20.9814i 0.947436 0.855136i
603603 0 0
604604 14.1498 + 1.45287i 0.575748 + 0.0591166i
605605 30.4982i 1.23993i
606606 0 0
607607 2.14863i 0.0872103i −0.999049 0.0436052i 0.986116π-0.986116\pi
0.999049 0.0436052i 0.0138844π-0.0138844\pi
608608 6.26672 + 10.7054i 0.254149 + 0.434161i
609609 0 0
610610 3.87838 + 4.29699i 0.157031 + 0.173980i
611611 −0.716953 −0.0290048
612612 0 0
613613 −44.8432 −1.81120 −0.905601 0.424131i 0.860579π-0.860579\pi
−0.905601 + 0.424131i 0.860579π0.860579\pi
614614 3.75844 + 4.16411i 0.151678 + 0.168050i
615615 0 0
616616 −61.3092 + 44.8819i −2.47022 + 1.80835i
617617 34.4459i 1.38674i −0.720581 0.693371i 0.756125π-0.756125\pi
0.720581 0.693371i 0.243875π-0.243875\pi
618618 0 0
619619 12.7731i 0.513396i 0.966492 + 0.256698i 0.0826345π0.0826345\pi
−0.966492 + 0.256698i 0.917365π0.917365\pi
620620 0.580662 5.65518i 0.0233199 0.227118i
621621 0 0
622622 20.3500 18.3675i 0.815961 0.736469i
623623 −25.5996 −1.02563
624624 0 0
625625 1.00000 0.0400000
626626 −23.0650 + 20.8180i −0.921862 + 0.832054i
627627 0 0
628628 −0.645275 + 6.28446i −0.0257493 + 0.250777i
629629 5.54817i 0.221220i
630630 0 0
631631 12.9765i 0.516586i −0.966067 0.258293i 0.916840π-0.916840\pi
0.966067 0.258293i 0.0831600π-0.0831600\pi
632632 0.565083 0.413674i 0.0224778 0.0164551i
633633 0 0
634634 25.2265 + 27.9493i 1.00187 + 1.11001i
635635 −2.16321 −0.0858445
636636 0 0
637637 −18.4388 −0.730570
638638 39.8415 + 44.1419i 1.57734 + 1.74759i
639639 0 0
640640 −10.2966 + 4.68837i −0.407007 + 0.185324i
641641 30.5564i 1.20691i −0.797398 0.603453i 0.793791π-0.793791\pi
0.797398 0.603453i 0.206209π-0.206209\pi
642642 0 0
643643 6.17425i 0.243489i 0.992561 + 0.121744i 0.0388488π0.0388488\pi
−0.992561 + 0.121744i 0.961151π0.961151\pi
644644 16.7603 + 1.72091i 0.660447 + 0.0678133i
645645 0 0
646646 2.18860 1.97538i 0.0861091 0.0777204i
647647 −9.68580 −0.380788 −0.190394 0.981708i 0.560977π-0.560977\pi
−0.190394 + 0.981708i 0.560977π0.560977\pi
648648 0 0
649649 17.3148 0.679663
650650 1.86310 1.68160i 0.0730769 0.0659577i
651651 0 0
652652 28.0649 + 2.88164i 1.09911 + 0.112854i
653653 16.3465i 0.639688i −0.947470 0.319844i 0.896369π-0.896369\pi
0.947470 0.319844i 0.103631π-0.103631\pi
654654 0 0
655655 1.91441i 0.0748023i
656656 −5.78387 + 27.8682i −0.225822 + 1.08807i
657657 0 0
658658 1.59634 + 1.76864i 0.0622319 + 0.0689490i
659659 −23.0858 −0.899295 −0.449648 0.893206i 0.648450π-0.648450\pi
−0.449648 + 0.893206i 0.648450π0.648450\pi
660660 0 0
661661 18.4701 0.718403 0.359202 0.933260i 0.383049π-0.383049\pi
0.359202 + 0.933260i 0.383049π0.383049\pi
662662 31.4746 + 34.8718i 1.22329 + 1.35533i
663663 0 0
664664 19.5022 + 26.6402i 0.756832 + 1.03384i
665665 9.14452i 0.354609i
666666 0 0
667667 13.1855i 0.510545i
668668 −3.74505 + 36.4738i −0.144900 + 1.41121i
669669 0 0
670670 −11.4389 + 10.3245i −0.441922 + 0.398870i
671671 26.3670 1.01789
672672 0 0
673673 −30.3154 −1.16857 −0.584286 0.811548i 0.698625π-0.698625\pi
−0.584286 + 0.811548i 0.698625π0.698625\pi
674674 −1.69775 + 1.53235i −0.0653948 + 0.0590240i
675675 0 0
676676 −2.01228 + 19.5980i −0.0773956 + 0.753771i
677677 42.3891i 1.62915i 0.580061 + 0.814573i 0.303029π0.303029\pi
−0.580061 + 0.814573i 0.696971π0.696971\pi
678678 0 0
679679 75.7715i 2.90784i
680680 1.58834 + 2.16969i 0.0609100 + 0.0832037i
681681 0 0
682682 −17.3505 19.2233i −0.664386 0.736097i
683683 20.7847 0.795303 0.397652 0.917536i 0.369825π-0.369825\pi
0.397652 + 0.917536i 0.369825π0.369825\pi
684684 0 0
685685 11.6608 0.445536
686686 13.3952 + 14.8410i 0.511429 + 0.566631i
687687 0 0
688688 4.31611 20.7961i 0.164550 0.792845i
689689 8.65764i 0.329830i
690690 0 0
691691 42.6767i 1.62350i −0.584007 0.811748i 0.698516π-0.698516\pi
0.584007 0.811748i 0.301484π-0.301484\pi
692692 15.6811 + 1.61010i 0.596105 + 0.0612068i
693693 0 0
694694 −5.44437 + 4.91398i −0.206665 + 0.186532i
695695 7.66107 0.290601
696696 0 0
697697 6.76460 0.256227
698698 19.2479 17.3728i 0.728544 0.657569i
699699 0 0
700700 −8.29663 0.851880i −0.313583 0.0321981i
701701 26.5424i 1.00249i −0.865305 0.501247i 0.832875π-0.832875\pi
0.865305 0.501247i 0.167125π-0.167125\pi
702702 0 0
703703 12.7975i 0.482668i
704704 −15.5719 + 49.1264i −0.586888 + 1.85152i
705705 0 0
706706 10.1364 + 11.2305i 0.381489 + 0.422665i
707707 −42.7016 −1.60596
708708 0 0
709709 46.2540 1.73711 0.868553 0.495596i 0.165050π-0.165050\pi
0.868553 + 0.495596i 0.165050π0.165050\pi
710710 −2.11424 2.34244i −0.0793461 0.0879104i
711711 0 0
712712 −14.0103 + 10.2563i −0.525057 + 0.384372i
713713 5.74213i 0.215045i
714714 0 0
715715 11.4323i 0.427543i
716716 −2.01035 + 19.5792i −0.0751304 + 0.731710i
717717 0 0
718718 −15.9530 + 14.3989i −0.595361 + 0.537361i
719719 44.9992 1.67819 0.839094 0.543986i 0.183086π-0.183086\pi
0.839094 + 0.543986i 0.183086π0.183086\pi
720720 0 0
721721 40.2828 1.50021
722722 −14.8985 + 13.4471i −0.554464 + 0.500448i
723723 0 0
724724 3.74275 36.4514i 0.139098 1.35470i
725725 6.52706i 0.242409i
726726 0 0
727727 35.6578i 1.32247i 0.750177 + 0.661237i 0.229968π0.229968\pi
−0.750177 + 0.661237i 0.770032π0.770032\pi
728728 −16.8900 + 12.3645i −0.625985 + 0.458257i
729729 0 0
730730 −1.65501 1.83365i −0.0612548 0.0678664i
731731 −5.04795 −0.186705
732732 0 0
733733 −47.1392 −1.74113 −0.870563 0.492057i 0.836245π-0.836245\pi
−0.870563 + 0.492057i 0.836245π0.836245\pi
734734 −23.2702 25.7819i −0.858920 0.951628i
735735 0 0
736736 9.86211 5.77308i 0.363522 0.212798i
737737 70.1906i 2.58551i
738738 0 0
739739 40.9282i 1.50557i −0.658268 0.752784i 0.728711π-0.728711\pi
0.658268 0.752784i 0.271289π-0.271289\pi
740740 11.6109 + 1.19219i 0.426826 + 0.0438256i
741741 0 0
742742 21.3574 19.2768i 0.784056 0.707673i
743743 −33.8553 −1.24203 −0.621014 0.783799i 0.713279π-0.713279\pi
−0.621014 + 0.783799i 0.713279π0.713279\pi
744744 0 0
745745 −3.49438 −0.128024
746746 −19.5363 + 17.6331i −0.715274 + 0.645592i
747747 0 0
748748 12.1843 + 1.25106i 0.445504 + 0.0457434i
749749 13.4844i 0.492708i
750750 0 0
751751 6.85461i 0.250128i 0.992149 + 0.125064i 0.0399137π0.0399137\pi
−0.992149 + 0.125064i 0.960086π0.960086\pi
752752 1.58225 + 0.328386i 0.0576987 + 0.0119750i
753753 0 0
754754 10.9759 + 12.1606i 0.399718 + 0.442862i
755755 7.11211 0.258836
756756 0 0
757757 40.8568 1.48497 0.742483 0.669865i 0.233648π-0.233648\pi
0.742483 + 0.669865i 0.233648π0.233648\pi
758758 −26.4417 29.2957i −0.960404 1.06407i
759759 0 0
760760 3.66370 + 5.00465i 0.132896 + 0.181538i
761761 27.9013i 1.01142i −0.862703 0.505710i 0.831231π-0.831231\pi
0.862703 0.505710i 0.168769π-0.168769\pi
762762 0 0
763763 12.0246i 0.435319i
764764 −3.89779 + 37.9614i −0.141017 + 1.37339i
765765 0 0
766766 −7.77118 + 7.01411i −0.280784 + 0.253430i
767767 4.77002 0.172235
768768 0 0
769769 6.12340 0.220816 0.110408 0.993886i 0.464784π-0.464784\pi
0.110408 + 0.993886i 0.464784π0.464784\pi
770770 −28.2022 + 25.4547i −1.01634 + 0.917323i
771771 0 0
772772 −2.65629 + 25.8702i −0.0956020 + 0.931087i
773773 14.8193i 0.533012i −0.963833 0.266506i 0.914131π-0.914131\pi
0.963833 0.266506i 0.0858692π-0.0858692\pi
774774 0 0
775775 2.84246i 0.102104i
776776 30.3574 + 41.4685i 1.08977 + 1.48863i
777777 0 0
778778 −16.9988 18.8336i −0.609438 0.675218i
779779 15.6034 0.559049
780780 0 0
781781 −14.3736 −0.514328
782782 −1.81978 2.01620i −0.0650751 0.0720991i
783783 0 0
784784 40.6927 + 8.44551i 1.45331 + 0.301625i
785785 3.15875i 0.112741i
786786 0 0
787787 15.7271i 0.560609i 0.959911 + 0.280305i 0.0904355π0.0904355\pi
−0.959911 + 0.280305i 0.909564π0.909564\pi
788788 −31.5266 3.23708i −1.12309 0.115316i
789789 0 0
790790 0.259938 0.234614i 0.00924816 0.00834720i
791791 −12.8783 −0.457900
792792 0 0
793793 7.26380 0.257945
794794 29.3008 26.4463i 1.03985 0.938544i
795795 0 0
796796 2.47822 + 0.254459i 0.0878383 + 0.00901904i
797797 43.3678i 1.53617i −0.640350 0.768083i 0.721211π-0.721211\pi
0.640350 0.768083i 0.278789π-0.278789\pi
798798 0 0
799799 0.384068i 0.0135873i
800800 −4.88192 + 2.85778i −0.172602 + 0.101038i
801801 0 0
802802 −3.78742 4.19622i −0.133738 0.148174i
803803 −11.2515 −0.397058
804804 0 0
805805 8.42420 0.296914
806806 −4.77987 5.29579i −0.168364 0.186536i
807807 0 0
808808 −23.3699 + 17.1081i −0.822150 + 0.601862i
809809 0.395982i 0.0139220i −0.999976 0.00696100i 0.997784π-0.997784\pi
0.999976 0.00696100i 0.00221577π-0.00221577\pi
810810 0 0
811811 45.9241i 1.61261i 0.591498 + 0.806307i 0.298537π0.298537\pi
−0.591498 + 0.806307i 0.701463π0.701463\pi
812812 5.56027 54.1526i 0.195127 1.90038i
813813 0 0
814814 39.4682 35.6232i 1.38336 1.24859i
815815 14.1062 0.494119
816816 0 0
817817 −11.6437 −0.407362
818818 −23.2318 + 20.9685i −0.812279 + 0.733147i
819819 0 0
820820 −1.45357 + 14.1566i −0.0507609 + 0.494370i
821821 34.0437i 1.18813i −0.804415 0.594067i 0.797521π-0.797521\pi
0.804415 0.594067i 0.202479π-0.202479\pi
822822 0 0
823823 10.5134i 0.366475i −0.983069 0.183238i 0.941342π-0.941342\pi
0.983069 0.183238i 0.0586578π-0.0586578\pi
824824 22.0462 16.1391i 0.768014 0.562231i
825825 0 0
826826 −10.6207 11.7671i −0.369543 0.409430i
827827 −40.8826 −1.42163 −0.710813 0.703381i 0.751673π-0.751673\pi
−0.710813 + 0.703381i 0.751673π0.751673\pi
828828 0 0
829829 24.0184 0.834194 0.417097 0.908862i 0.363048π-0.363048\pi
0.417097 + 0.908862i 0.363048π0.363048\pi
830830 11.0606 + 12.2545i 0.383920 + 0.425359i
831831 0 0
832832 −4.28988 + 13.5337i −0.148725 + 0.469198i
833833 9.87754i 0.342237i
834834 0 0
835835 18.3328i 0.634432i
836836 28.1047 + 2.88573i 0.972021 + 0.0998050i
837837 0 0
838838 33.1890 29.9557i 1.14649 1.03480i
839839 −34.1904 −1.18038 −0.590192 0.807263i 0.700948π-0.700948\pi
−0.590192 + 0.807263i 0.700948π0.700948\pi
840840 0 0
841841 −13.6025 −0.469052
842842 −31.5543 + 28.4803i −1.08743 + 0.981495i
843843 0 0
844844 −32.4781 3.33478i −1.11794 0.114788i
845845 9.85054i 0.338869i
846846 0 0
847847 127.181i 4.37000i
848848 3.96546 19.1066i 0.136174 0.656124i
849849 0 0
850850 0.900823 + 0.998053i 0.0308980 + 0.0342329i
851851 −11.7895 −0.404137
852852 0 0
853853 −24.7483 −0.847366 −0.423683 0.905811i 0.639263π-0.639263\pi
−0.423683 + 0.905811i 0.639263π0.639263\pi
854854 −16.1733 17.9190i −0.553440 0.613176i
855855 0 0
856856 5.40243 + 7.37979i 0.184651 + 0.252236i
857857 10.6711i 0.364518i 0.983251 + 0.182259i 0.0583410π0.0583410\pi
−0.983251 + 0.182259i 0.941659π0.941659\pi
858858 0 0
859859 23.8104i 0.812401i 0.913784 + 0.406200i 0.133146π0.133146\pi
−0.913784 + 0.406200i 0.866854π0.866854\pi
860860 1.08470 10.5641i 0.0369880 0.360233i
861861 0 0
862862 26.3331 23.7677i 0.896909 0.809532i
863863 −53.1072 −1.80779 −0.903896 0.427753i 0.859305π-0.859305\pi
−0.903896 + 0.427753i 0.859305π0.859305\pi
864864 0 0
865865 7.88176 0.267988
866866 −30.8962 + 27.8863i −1.04989 + 0.947614i
867867 0 0
868868 −2.42143 + 23.5828i −0.0821888 + 0.800454i
869869 1.59502i 0.0541072i
870870 0 0
871871 19.3367i 0.655200i
872872 4.81758 + 6.58087i 0.163144 + 0.222856i
873873 0 0
874874 −4.19754 4.65061i −0.141984 0.157309i
875875 −4.17013 −0.140976
876876 0 0
877877 −41.4276 −1.39891 −0.699456 0.714676i 0.746574π-0.746574\pi
−0.699456 + 0.714676i 0.746574π0.746574\pi
878878 −27.1524 30.0831i −0.916348 1.01525i
879879 0 0
880880 −5.23632 + 25.2300i −0.176516 + 0.850502i
881881 18.8579i 0.635340i −0.948201 0.317670i 0.897100π-0.897100\pi
0.948201 0.317670i 0.102900π-0.102900\pi
882882 0 0
883883 36.0305i 1.21252i −0.795266 0.606261i 0.792669π-0.792669\pi
0.795266 0.606261i 0.207331π-0.207331\pi
884884 3.35665 + 0.344653i 0.112896 + 0.0115919i
885885 0 0
886886 −31.3457 + 28.2920i −1.05308 + 0.950489i
887887 −19.3362 −0.649245 −0.324623 0.945844i 0.605237π-0.605237\pi
−0.324623 + 0.945844i 0.605237π0.605237\pi
888888 0 0
889889 9.02088 0.302551
890890 −6.44470 + 5.81686i −0.216027 + 0.194981i
891891 0 0
892892 −35.6640 3.66190i −1.19412 0.122609i
893893 0.885900i 0.0296455i
894894 0 0
895895 9.84108i 0.328951i
896896 42.9380 19.5511i 1.43446 0.653157i
897897 0 0
898898 13.9306 + 15.4343i 0.464872 + 0.515048i
899899 18.5529 0.618774
900900 0 0
901901 −4.63785 −0.154509
902902 43.4336 + 48.1216i 1.44618 + 1.60227i
903903 0 0
904904 −7.04809 + 5.15961i −0.234416 + 0.171606i
905905 18.3215i 0.609027i
906906 0 0
907907 17.1102i 0.568134i −0.958804 0.284067i 0.908316π-0.908316\pi
0.958804 0.284067i 0.0916838π-0.0916838\pi
908908 2.13075 20.7518i 0.0707114 0.688672i
909909 0 0
910910 −7.76937 + 7.01248i −0.257552 + 0.232461i
911911 −11.3283 −0.375324 −0.187662 0.982234i 0.560091π-0.560091\pi
−0.187662 + 0.982234i 0.560091π0.560091\pi
912912 0 0
913913 75.1952 2.48860
914914 9.27352 8.37009i 0.306741 0.276858i
915915 0 0
916916 −3.83799 + 37.3790i −0.126811 + 1.23504i
917917 7.98334i 0.263633i
918918 0 0
919919 33.3567i 1.10034i 0.835054 + 0.550169i 0.185437π0.185437\pi
−0.835054 + 0.550169i 0.814563π0.814563\pi
920920 4.61043 3.37510i 0.152001 0.111274i
921921 0 0
922922 0.865470 + 0.958885i 0.0285027 + 0.0315792i
923923 −3.95976 −0.130337
924924 0 0
925925 5.83599 0.191886
926926 7.45853 + 8.26357i 0.245102 + 0.271558i
927927 0 0
928928 −18.6529 31.8646i −0.612311 1.04601i
929929 10.4925i 0.344248i 0.985075 + 0.172124i 0.0550630π0.0550630\pi
−0.985075 + 0.172124i 0.944937π0.944937\pi
930930 0 0
931931 22.7838i 0.746708i
932932 −14.4976 1.48859i −0.474886 0.0487602i
933933 0 0
934934 30.1054 27.1725i 0.985079 0.889113i
935935 6.12420 0.200283
936936 0 0
937937 29.4458 0.961952 0.480976 0.876734i 0.340282π-0.340282\pi
0.480976 + 0.876734i 0.340282π0.340282\pi
938938 47.7016 43.0545i 1.55751 1.40578i
939939 0 0
940940 0.803758 + 0.0825282i 0.0262157 + 0.00269177i
941941 36.0415i 1.17492i −0.809254 0.587459i 0.800128π-0.800128\pi
0.809254 0.587459i 0.199872π-0.199872\pi
942942 0 0
943943 14.3743i 0.468091i
944944 −10.5270 2.18481i −0.342624 0.0711095i
945945 0 0
946946 −32.4115 35.9098i −1.05379 1.16753i
947947 −12.1726 −0.395557 −0.197778 0.980247i 0.563373π-0.563373\pi
−0.197778 + 0.980247i 0.563373π0.563373\pi
948948 0 0
949949 −3.09967 −0.100620
950950 2.07786 + 2.30213i 0.0674146 + 0.0746911i
951951 0 0
952952 −6.62357 9.04787i −0.214671 0.293243i
953953 25.0300i 0.810801i −0.914139 0.405400i 0.867132π-0.867132\pi
0.914139 0.405400i 0.132868π-0.132868\pi
954954 0 0
955955 19.0805i 0.617430i
956956 6.06342 59.0528i 0.196105 1.90991i
957957 0 0
958958 −27.6963 + 24.9981i −0.894827 + 0.807653i
959959 −48.6270 −1.57025
960960 0 0
961961 22.9204 0.739369
962962 10.8730 9.81379i 0.350561 0.316409i
963963 0 0
964964 −2.60466 + 25.3673i −0.0838905 + 0.817026i
965965 13.0031i 0.418584i
966966 0 0
967967 2.75536i 0.0886063i −0.999018 0.0443032i 0.985893π-0.985893\pi
0.999018 0.0443032i 0.0141068π-0.0141068\pi
968968 50.9544 + 69.6043i 1.63774 + 2.23717i
969969 0 0
970970 17.2171 + 19.0755i 0.552809 + 0.612476i
971971 33.6700 1.08052 0.540261 0.841497i 0.318325π-0.318325\pi
0.540261 + 0.841497i 0.318325π0.318325\pi
972972 0 0
973973 −31.9476 −1.02419
974974 7.45936 + 8.26448i 0.239013 + 0.264811i
975975 0 0
976976 −16.0305 3.32704i −0.513125 0.106496i
977977 56.0751i 1.79400i 0.442030 + 0.897000i 0.354258π0.354258\pi
−0.442030 + 0.897000i 0.645742π0.645742\pi
978978 0 0
979979 39.5457i 1.26388i
980980 20.6712 + 2.12248i 0.660318 + 0.0678001i
981981 0 0
982982 20.4654 18.4716i 0.653076 0.589453i
983983 −15.0532 −0.480124 −0.240062 0.970758i 0.577168π-0.577168\pi
−0.240062 + 0.970758i 0.577168π0.577168\pi
984984 0 0
985985 −15.8462 −0.504901
986986 −6.51435 + 5.87972i −0.207459 + 0.187249i
987987 0 0
988988 7.74252 + 0.794985i 0.246322 + 0.0252919i
989989 10.7265i 0.341084i
990990 0 0
991991 32.4096i 1.02953i −0.857333 0.514763i 0.827880π-0.827880\pi
0.857333 0.514763i 0.172120π-0.172120\pi
992992 8.12311 + 13.8766i 0.257909 + 0.440584i
993993 0 0
994994 8.81666 + 9.76829i 0.279648 + 0.309831i
995995 1.24563 0.0394890
996996 0 0
997997 −44.1499 −1.39824 −0.699120 0.715004i 0.746425π-0.746425\pi
−0.699120 + 0.715004i 0.746425π0.746425\pi
998998 0.802069 + 0.888641i 0.0253891 + 0.0281294i
999999 0 0
Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1620.2.e.b.971.12 48
3.2 odd 2 inner 1620.2.e.b.971.37 48
4.3 odd 2 inner 1620.2.e.b.971.38 48
9.2 odd 6 180.2.q.a.131.3 yes 48
9.4 even 3 180.2.q.a.11.10 yes 48
9.5 odd 6 540.2.q.a.251.15 48
9.7 even 3 540.2.q.a.71.22 48
12.11 even 2 inner 1620.2.e.b.971.11 48
36.7 odd 6 540.2.q.a.71.15 48
36.11 even 6 180.2.q.a.131.10 yes 48
36.23 even 6 540.2.q.a.251.22 48
36.31 odd 6 180.2.q.a.11.3 48
45.2 even 12 900.2.o.b.599.15 48
45.4 even 6 900.2.r.f.551.15 48
45.13 odd 12 900.2.o.b.299.2 48
45.22 odd 12 900.2.o.c.299.23 48
45.29 odd 6 900.2.r.f.851.22 48
45.38 even 12 900.2.o.c.599.10 48
180.47 odd 12 900.2.o.b.599.2 48
180.67 even 12 900.2.o.c.299.10 48
180.83 odd 12 900.2.o.c.599.23 48
180.103 even 12 900.2.o.b.299.15 48
180.119 even 6 900.2.r.f.851.15 48
180.139 odd 6 900.2.r.f.551.22 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.2.q.a.11.3 48 36.31 odd 6
180.2.q.a.11.10 yes 48 9.4 even 3
180.2.q.a.131.3 yes 48 9.2 odd 6
180.2.q.a.131.10 yes 48 36.11 even 6
540.2.q.a.71.15 48 36.7 odd 6
540.2.q.a.71.22 48 9.7 even 3
540.2.q.a.251.15 48 9.5 odd 6
540.2.q.a.251.22 48 36.23 even 6
900.2.o.b.299.2 48 45.13 odd 12
900.2.o.b.299.15 48 180.103 even 12
900.2.o.b.599.2 48 180.47 odd 12
900.2.o.b.599.15 48 45.2 even 12
900.2.o.c.299.10 48 180.67 even 12
900.2.o.c.299.23 48 45.22 odd 12
900.2.o.c.599.10 48 45.38 even 12
900.2.o.c.599.23 48 180.83 odd 12
900.2.r.f.551.15 48 45.4 even 6
900.2.r.f.551.22 48 180.139 odd 6
900.2.r.f.851.15 48 180.119 even 6
900.2.r.f.851.22 48 45.29 odd 6
1620.2.e.b.971.11 48 12.11 even 2 inner
1620.2.e.b.971.12 48 1.1 even 1 trivial
1620.2.e.b.971.37 48 3.2 odd 2 inner
1620.2.e.b.971.38 48 4.3 odd 2 inner