Properties

Label 792.2.f.g.397.10
Level 792792
Weight 22
Character 792.397
Analytic conductor 6.3246.324
Analytic rank 00
Dimension 1010
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [792,2,Mod(397,792)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(792, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("792.397"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 792=233211 792 = 2^{3} \cdot 3^{2} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 792.f (of order 22, degree 11, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-4,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 6.324151840096.32415184009
Analytic rank: 00
Dimension: 1010
Coefficient field: 10.0.578281160704.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x10+2x82x73x66x56x48x3+16x2+32 x^{10} + 2x^{8} - 2x^{7} - 3x^{6} - 6x^{5} - 6x^{4} - 8x^{3} + 16x^{2} + 32 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 397.10
Root 1.413630.0406696i1.41363 - 0.0406696i of defining polynomial
Character χ\chi == 792.397
Dual form 792.2.f.g.397.9

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
f(q)f(q) == q+(1.29865+0.559929i)q2+(1.37296+1.45430i)q42.51595iq5+1.47743q7+(0.968683+2.65738i)q8+(1.408753.26733i)q10+1.00000iq11+3.47743iq13+(1.91866+0.827257i)q14+(0.229967+3.99338i)q16+3.31475q177.13195iq19+(3.658943.45430i)q20+(0.559929+1.29865i)q22+6.45332q231.33001q25+(1.94712+4.51595i)q26+(2.02845+2.14863i)q28+1.41480iq29+0.636125q31+(2.53466+5.05722i)q32+(4.30469+1.85603i)q343.71715iq356.97588iq37+(3.993389.26187i)q38+(6.685832.43716i)q406.72955q413.21471iq43+(1.45430+1.37296i)q44+(8.38057+3.61340i)q460.862328q474.81719q49+(1.727210.744712i)q50+(5.05722+4.77437i)q52+13.2515iq53+2.51595q55+(1.43116+3.92610i)q56+(0.792187+1.83732i)q582.63236iq59+6.45993iq61+(0.826100+0.356185i)q62+(6.12331+5.14831i)q64+8.74905q65+7.66426iq67+(4.55102+4.82064i)q68+(2.081344.82726i)q7012.2900q7113.2440q73+(3.906009.05920i)q74+(10.37209.79187i)q76+1.47743iq7716.3409q79+(10.0472+0.578585i)q80+(8.739303.76807i)q8213.8040iq838.33976iq85+(1.800014.17477i)q86+(2.65738+0.968683i)q88+1.04979q89+5.13767iq91+(8.86014+9.38505i)q92+(1.119860.482842i)q9417.9436q95+16.2110q97+(6.255822.69729i)q98+O(q100)q+(1.29865 + 0.559929i) q^{2} +(1.37296 + 1.45430i) q^{4} -2.51595i q^{5} +1.47743 q^{7} +(0.968683 + 2.65738i) q^{8} +(1.40875 - 3.26733i) q^{10} +1.00000i q^{11} +3.47743i q^{13} +(1.91866 + 0.827257i) q^{14} +(-0.229967 + 3.99338i) q^{16} +3.31475 q^{17} -7.13195i q^{19} +(3.65894 - 3.45430i) q^{20} +(-0.559929 + 1.29865i) q^{22} +6.45332 q^{23} -1.33001 q^{25} +(-1.94712 + 4.51595i) q^{26} +(2.02845 + 2.14863i) q^{28} +1.41480i q^{29} +0.636125 q^{31} +(-2.53466 + 5.05722i) q^{32} +(4.30469 + 1.85603i) q^{34} -3.71715i q^{35} -6.97588i q^{37} +(3.99338 - 9.26187i) q^{38} +(6.68583 - 2.43716i) q^{40} -6.72955 q^{41} -3.21471i q^{43} +(-1.45430 + 1.37296i) q^{44} +(8.38057 + 3.61340i) q^{46} -0.862328 q^{47} -4.81719 q^{49} +(-1.72721 - 0.744712i) q^{50} +(-5.05722 + 4.77437i) q^{52} +13.2515i q^{53} +2.51595 q^{55} +(1.43116 + 3.92610i) q^{56} +(-0.792187 + 1.83732i) q^{58} -2.63236i q^{59} +6.45993i q^{61} +(0.826100 + 0.356185i) q^{62} +(-6.12331 + 5.14831i) q^{64} +8.74905 q^{65} +7.66426i q^{67} +(4.55102 + 4.82064i) q^{68} +(2.08134 - 4.82726i) q^{70} -12.2900 q^{71} -13.2440 q^{73} +(3.90600 - 9.05920i) q^{74} +(10.3720 - 9.79187i) q^{76} +1.47743i q^{77} -16.3409 q^{79} +(10.0472 + 0.578585i) q^{80} +(-8.73930 - 3.76807i) q^{82} -13.8040i q^{83} -8.33976i q^{85} +(1.80001 - 4.17477i) q^{86} +(-2.65738 + 0.968683i) q^{88} +1.04979 q^{89} +5.13767i q^{91} +(8.86014 + 9.38505i) q^{92} +(-1.11986 - 0.482842i) q^{94} -17.9436 q^{95} +16.2110 q^{97} +(-6.25582 - 2.69729i) q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q4q4+10q10+12q14+4q1712q20+12q236q25+20q2612q284q31+20q328q38+20q404q414q44+2q46+4q476q49+24q98+O(q100) 10 q - 4 q^{4} + 10 q^{10} + 12 q^{14} + 4 q^{17} - 12 q^{20} + 12 q^{23} - 6 q^{25} + 20 q^{26} - 12 q^{28} - 4 q^{31} + 20 q^{32} - 8 q^{38} + 20 q^{40} - 4 q^{41} - 4 q^{44} + 2 q^{46} + 4 q^{47} - 6 q^{49}+ \cdots - 24 q^{98}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/792Z)×\left(\mathbb{Z}/792\mathbb{Z}\right)^\times.

nn 145145 199199 353353 397397
χ(n)\chi(n) 11 11 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 1.29865 + 0.559929i 0.918281 + 0.395930i
33 0 0
44 1.37296 + 1.45430i 0.686480 + 0.727149i
55 2.51595i 1.12517i −0.826740 0.562584i 0.809807π-0.809807\pi
0.826740 0.562584i 0.190193π-0.190193\pi
66 0 0
77 1.47743 0.558417 0.279209 0.960230i 0.409928π-0.409928\pi
0.279209 + 0.960230i 0.409928π0.409928\pi
88 0.968683 + 2.65738i 0.342481 + 0.939525i
99 0 0
1010 1.40875 3.26733i 0.445487 1.03322i
1111 1.00000i 0.301511i
1212 0 0
1313 3.47743i 0.964466i 0.876043 + 0.482233i 0.160174π0.160174\pi
−0.876043 + 0.482233i 0.839826π0.839826\pi
1414 1.91866 + 0.827257i 0.512784 + 0.221094i
1515 0 0
1616 −0.229967 + 3.99338i −0.0574917 + 0.998346i
1717 3.31475 0.803946 0.401973 0.915652i 0.368325π-0.368325\pi
0.401973 + 0.915652i 0.368325π0.368325\pi
1818 0 0
1919 7.13195i 1.63618i −0.575090 0.818090i 0.695033π-0.695033\pi
0.575090 0.818090i 0.304967π-0.304967\pi
2020 3.65894 3.45430i 0.818165 0.772405i
2121 0 0
2222 −0.559929 + 1.29865i −0.119377 + 0.276872i
2323 6.45332 1.34561 0.672805 0.739820i 0.265089π-0.265089\pi
0.672805 + 0.739820i 0.265089π0.265089\pi
2424 0 0
2525 −1.33001 −0.266002
2626 −1.94712 + 4.51595i −0.381861 + 0.885651i
2727 0 0
2828 2.02845 + 2.14863i 0.383342 + 0.406052i
2929 1.41480i 0.262722i 0.991335 + 0.131361i 0.0419346π0.0419346\pi
−0.991335 + 0.131361i 0.958065π0.958065\pi
3030 0 0
3131 0.636125 0.114251 0.0571257 0.998367i 0.481806π-0.481806\pi
0.0571257 + 0.998367i 0.481806π0.481806\pi
3232 −2.53466 + 5.05722i −0.448068 + 0.893999i
3333 0 0
3434 4.30469 + 1.85603i 0.738248 + 0.318306i
3535 3.71715i 0.628313i
3636 0 0
3737 6.97588i 1.14683i −0.819266 0.573414i 0.805619π-0.805619\pi
0.819266 0.573414i 0.194381π-0.194381\pi
3838 3.99338 9.26187i 0.647812 1.50247i
3939 0 0
4040 6.68583 2.43716i 1.05712 0.385349i
4141 −6.72955 −1.05098 −0.525490 0.850800i 0.676118π-0.676118\pi
−0.525490 + 0.850800i 0.676118π0.676118\pi
4242 0 0
4343 3.21471i 0.490239i −0.969493 0.245119i 0.921173π-0.921173\pi
0.969493 0.245119i 0.0788271π-0.0788271\pi
4444 −1.45430 + 1.37296i −0.219244 + 0.206981i
4545 0 0
4646 8.38057 + 3.61340i 1.23565 + 0.532767i
4747 −0.862328 −0.125783 −0.0628917 0.998020i 0.520032π-0.520032\pi
−0.0628917 + 0.998020i 0.520032π0.520032\pi
4848 0 0
4949 −4.81719 −0.688170
5050 −1.72721 0.744712i −0.244265 0.105318i
5151 0 0
5252 −5.05722 + 4.77437i −0.701311 + 0.662086i
5353 13.2515i 1.82023i 0.414353 + 0.910116i 0.364008π0.364008\pi
−0.414353 + 0.910116i 0.635992π0.635992\pi
5454 0 0
5555 2.51595 0.339251
5656 1.43116 + 3.92610i 0.191247 + 0.524647i
5757 0 0
5858 −0.792187 + 1.83732i −0.104019 + 0.241252i
5959 2.63236i 0.342704i −0.985210 0.171352i 0.945186π-0.945186\pi
0.985210 0.171352i 0.0548136π-0.0548136\pi
6060 0 0
6161 6.45993i 0.827110i 0.910479 + 0.413555i 0.135713π0.135713\pi
−0.910479 + 0.413555i 0.864287π0.864287\pi
6262 0.826100 + 0.356185i 0.104915 + 0.0452355i
6363 0 0
6464 −6.12331 + 5.14831i −0.765413 + 0.643539i
6565 8.74905 1.08519
6666 0 0
6767 7.66426i 0.936339i 0.883639 + 0.468169i 0.155086π0.155086\pi
−0.883639 + 0.468169i 0.844914π0.844914\pi
6868 4.55102 + 4.82064i 0.551892 + 0.584589i
6969 0 0
7070 2.08134 4.82726i 0.248768 0.576968i
7171 −12.2900 −1.45856 −0.729278 0.684218i 0.760144π-0.760144\pi
−0.729278 + 0.684218i 0.760144π0.760144\pi
7272 0 0
7373 −13.2440 −1.55009 −0.775045 0.631905i 0.782273π-0.782273\pi
−0.775045 + 0.631905i 0.782273π0.782273\pi
7474 3.90600 9.05920i 0.454063 1.05311i
7575 0 0
7676 10.3720 9.79187i 1.18975 1.12320i
7777 1.47743i 0.168369i
7878 0 0
7979 −16.3409 −1.83850 −0.919249 0.393676i 0.871203π-0.871203\pi
−0.919249 + 0.393676i 0.871203π0.871203\pi
8080 10.0472 + 0.578585i 1.12331 + 0.0646878i
8181 0 0
8282 −8.73930 3.76807i −0.965094 0.416114i
8383 13.8040i 1.51518i −0.652730 0.757591i 0.726376π-0.726376\pi
0.652730 0.757591i 0.273624π-0.273624\pi
8484 0 0
8585 8.33976i 0.904574i
8686 1.80001 4.17477i 0.194100 0.450177i
8787 0 0
8888 −2.65738 + 0.968683i −0.283277 + 0.103262i
8989 1.04979 0.111277 0.0556387 0.998451i 0.482280π-0.482280\pi
0.0556387 + 0.998451i 0.482280π0.482280\pi
9090 0 0
9191 5.13767i 0.538574i
9292 8.86014 + 9.38505i 0.923734 + 0.978459i
9393 0 0
9494 −1.11986 0.482842i −0.115505 0.0498014i
9595 −17.9436 −1.84098
9696 0 0
9797 16.2110 1.64598 0.822989 0.568057i 0.192305π-0.192305\pi
0.822989 + 0.568057i 0.192305π0.192305\pi
9898 −6.25582 2.69729i −0.631934 0.272467i
9999 0 0
100100 −1.82605 1.93423i −0.182605 0.193423i
101101 8.63460i 0.859175i 0.903025 + 0.429588i 0.141341π0.141341\pi
−0.903025 + 0.429588i 0.858659π0.858659\pi
102102 0 0
103103 −4.95487 −0.488217 −0.244109 0.969748i 0.578495π-0.578495\pi
−0.244109 + 0.969748i 0.578495π0.578495\pi
104104 −9.24085 + 3.36853i −0.906140 + 0.330312i
105105 0 0
106106 −7.41989 + 17.2090i −0.720684 + 1.67148i
107107 6.34666i 0.613554i −0.951781 0.306777i 0.900749π-0.900749\pi
0.951781 0.306777i 0.0992507π-0.0992507\pi
108108 0 0
109109 7.94246i 0.760750i 0.924832 + 0.380375i 0.124205π0.124205\pi
−0.924832 + 0.380375i 0.875795π0.875795\pi
110110 3.26733 + 1.40875i 0.311528 + 0.134319i
111111 0 0
112112 −0.339760 + 5.89996i −0.0321043 + 0.557493i
113113 −4.88248 −0.459305 −0.229653 0.973273i 0.573759π-0.573759\pi
−0.229653 + 0.973273i 0.573759π0.573759\pi
114114 0 0
115115 16.2362i 1.51404i
116116 −2.05754 + 1.94246i −0.191038 + 0.180353i
117117 0 0
118118 1.47394 3.41850i 0.135687 0.314699i
119119 4.89733 0.448937
120120 0 0
121121 −1.00000 −0.0909091
122122 −3.61710 + 8.38916i −0.327477 + 0.759519i
123123 0 0
124124 0.873373 + 0.925115i 0.0784312 + 0.0830778i
125125 9.23351i 0.825871i
126126 0 0
127127 −4.12017 −0.365606 −0.182803 0.983150i 0.558517π-0.558517\pi
−0.182803 + 0.983150i 0.558517π0.558517\pi
128128 −10.8347 + 3.25721i −0.957661 + 0.287900i
129129 0 0
130130 11.3619 + 4.89885i 0.996506 + 0.429657i
131131 13.3015i 1.16216i −0.813847 0.581080i 0.802630π-0.802630\pi
0.813847 0.581080i 0.197370π-0.197370\pi
132132 0 0
133133 10.5370i 0.913671i
134134 −4.29144 + 9.95316i −0.370724 + 0.859822i
135135 0 0
136136 3.21095 + 8.80855i 0.275336 + 0.755327i
137137 6.84444 0.584760 0.292380 0.956302i 0.405553π-0.405553\pi
0.292380 + 0.956302i 0.405553π0.405553\pi
138138 0 0
139139 12.1966i 1.03450i −0.855834 0.517250i 0.826956π-0.826956\pi
0.855834 0.517250i 0.173044π-0.173044\pi
140140 5.40584 5.10349i 0.456877 0.431324i
141141 0 0
142142 −15.9604 6.88153i −1.33936 0.577485i
143143 −3.47743 −0.290798
144144 0 0
145145 3.55956 0.295606
146146 −17.1992 7.41569i −1.42342 0.613727i
147147 0 0
148148 10.1450 9.57760i 0.833915 0.787274i
149149 3.04004i 0.249050i 0.992216 + 0.124525i 0.0397407π0.0397407\pi
−0.992216 + 0.124525i 0.960259π0.960259\pi
150150 0 0
151151 −1.83669 −0.149468 −0.0747339 0.997204i 0.523811π-0.523811\pi
−0.0747339 + 0.997204i 0.523811π0.523811\pi
152152 18.9523 6.90860i 1.53723 0.560361i
153153 0 0
154154 −0.827257 + 1.91866i −0.0666623 + 0.154610i
155155 1.60046i 0.128552i
156156 0 0
157157 10.0560i 0.802558i 0.915956 + 0.401279i 0.131434π0.131434\pi
−0.915956 + 0.401279i 0.868566π0.868566\pi
158158 −21.2211 9.14976i −1.68826 0.727916i
159159 0 0
160160 12.7237 + 6.37707i 1.00590 + 0.504152i
161161 9.53434 0.751411
162162 0 0
163163 12.8934i 1.00989i −0.863152 0.504945i 0.831513π-0.831513\pi
0.863152 0.504945i 0.168487π-0.168487\pi
164164 −9.23940 9.78678i −0.721476 0.764219i
165165 0 0
166166 7.72924 17.9264i 0.599905 1.39136i
167167 −4.47434 −0.346235 −0.173117 0.984901i 0.555384π-0.555384\pi
−0.173117 + 0.984901i 0.555384π0.555384\pi
168168 0 0
169169 0.907463 0.0698048
170170 4.66967 10.8304i 0.358148 0.830653i
171171 0 0
172172 4.67515 4.41366i 0.356477 0.336539i
173173 0.182807i 0.0138986i −0.999976 0.00694928i 0.997788π-0.997788\pi
0.999976 0.00694928i 0.00221204π-0.00221204\pi
174174 0 0
175175 −1.96500 −0.148540
176176 −3.99338 0.229967i −0.301013 0.0173344i
177177 0 0
178178 1.36330 + 0.587808i 0.102184 + 0.0440580i
179179 20.0915i 1.50171i 0.660469 + 0.750853i 0.270358π0.270358\pi
−0.660469 + 0.750853i 0.729642π0.729642\pi
180180 0 0
181181 7.63413i 0.567440i −0.958907 0.283720i 0.908431π-0.908431\pi
0.958907 0.283720i 0.0915686π-0.0915686\pi
182182 −2.87673 + 6.67201i −0.213238 + 0.494563i
183183 0 0
184184 6.25122 + 17.1489i 0.460846 + 1.26423i
185185 −17.5510 −1.29037
186186 0 0
187187 3.31475i 0.242399i
188188 −1.18394 1.25408i −0.0863478 0.0914633i
189189 0 0
190190 −23.3024 10.0472i −1.69053 0.728898i
191191 22.9943 1.66381 0.831906 0.554917i 0.187250π-0.187250\pi
0.831906 + 0.554917i 0.187250π0.187250\pi
192192 0 0
193193 0.735278 0.0529265 0.0264632 0.999650i 0.491576π-0.491576\pi
0.0264632 + 0.999650i 0.491576π0.491576\pi
194194 21.0524 + 9.07701i 1.51147 + 0.651692i
195195 0 0
196196 −6.61381 7.00564i −0.472415 0.500403i
197197 11.1665i 0.795579i −0.917477 0.397789i 0.869777π-0.869777\pi
0.917477 0.397789i 0.130223π-0.130223\pi
198198 0 0
199199 −18.1315 −1.28531 −0.642655 0.766156i 0.722167π-0.722167\pi
−0.642655 + 0.766156i 0.722167π0.722167\pi
200200 −1.28836 3.53434i −0.0911008 0.249916i
201201 0 0
202202 −4.83476 + 11.2133i −0.340173 + 0.788964i
203203 2.09027i 0.146708i
204204 0 0
205205 16.9312i 1.18253i
206206 −6.43461 2.77437i −0.448321 0.193300i
207207 0 0
208208 −13.8867 0.799694i −0.962871 0.0554488i
209209 7.13195 0.493327
210210 0 0
211211 18.4180i 1.26795i 0.773355 + 0.633973i 0.218577π0.218577\pi
−0.773355 + 0.633973i 0.781423π0.781423\pi
212212 −19.2716 + 18.1938i −1.32358 + 1.24955i
213213 0 0
214214 3.55368 8.24206i 0.242924 0.563415i
215215 −8.08805 −0.551601
216216 0 0
217217 0.939831 0.0637999
218218 −4.44721 + 10.3144i −0.301203 + 0.698582i
219219 0 0
220220 3.45430 + 3.65894i 0.232889 + 0.246686i
221221 11.5268i 0.775379i
222222 0 0
223223 14.5654 0.975368 0.487684 0.873020i 0.337842π-0.337842\pi
0.487684 + 0.873020i 0.337842π0.337842\pi
224224 −3.74478 + 7.47171i −0.250209 + 0.499224i
225225 0 0
226226 −6.34061 2.73384i −0.421771 0.181853i
227227 7.15145i 0.474658i 0.971429 + 0.237329i 0.0762720π0.0762720\pi
−0.971429 + 0.237329i 0.923728π0.923728\pi
228228 0 0
229229 6.48500i 0.428541i 0.976774 + 0.214271i 0.0687374π0.0687374\pi
−0.976774 + 0.214271i 0.931263π0.931263\pi
230230 9.09114 21.0851i 0.599452 1.39031i
231231 0 0
232232 −3.75965 + 1.37049i −0.246833 + 0.0899772i
233233 −11.4786 −0.751988 −0.375994 0.926622i 0.622699π-0.622699\pi
−0.375994 + 0.926622i 0.622699π0.622699\pi
234234 0 0
235235 2.16957i 0.141527i
236236 3.82824 3.61412i 0.249197 0.235259i
237237 0 0
238238 6.35989 + 2.74215i 0.412250 + 0.177747i
239239 −0.112036 −0.00724698 −0.00362349 0.999993i 0.501153π-0.501153\pi
−0.00362349 + 0.999993i 0.501153π0.501153\pi
240240 0 0
241241 2.93937 0.189341 0.0946706 0.995509i 0.469820π-0.469820\pi
0.0946706 + 0.995509i 0.469820π0.469820\pi
242242 −1.29865 0.559929i −0.0834801 0.0359936i
243243 0 0
244244 −9.39467 + 8.86922i −0.601432 + 0.567794i
245245 12.1198i 0.774307i
246246 0 0
247247 24.8009 1.57804
248248 0.616203 + 1.69042i 0.0391289 + 0.107342i
249249 0 0
250250 5.17011 11.9911i 0.326987 0.758381i
251251 31.2199i 1.97058i −0.170882 0.985291i 0.554662π-0.554662\pi
0.170882 0.985291i 0.445338π-0.445338\pi
252252 0 0
253253 6.45332i 0.405717i
254254 −5.35064 2.30700i −0.335729 0.144754i
255255 0 0
256256 −15.8942 1.83669i −0.993389 0.114793i
257257 −7.09741 −0.442725 −0.221362 0.975192i 0.571050π-0.571050\pi
−0.221362 + 0.975192i 0.571050π0.571050\pi
258258 0 0
259259 10.3064i 0.640409i
260260 12.0121 + 12.7237i 0.744958 + 0.789092i
261261 0 0
262262 7.44791 17.2740i 0.460133 1.06719i
263263 −16.3863 −1.01042 −0.505211 0.862996i 0.668585π-0.668585\pi
−0.505211 + 0.862996i 0.668585π0.668585\pi
264264 0 0
265265 33.3401 2.04807
266266 5.89996 13.6838i 0.361749 0.839007i
267267 0 0
268268 −11.1461 + 10.5227i −0.680858 + 0.642777i
269269 17.1262i 1.04420i −0.852883 0.522102i 0.825148π-0.825148\pi
0.852883 0.522102i 0.174852π-0.174852\pi
270270 0 0
271271 19.5142 1.18540 0.592702 0.805422i 0.298061π-0.298061\pi
0.592702 + 0.805422i 0.298061π0.298061\pi
272272 −0.762283 + 13.2371i −0.0462202 + 0.802616i
273273 0 0
274274 8.88850 + 3.83240i 0.536974 + 0.231524i
275275 1.33001i 0.0802027i
276276 0 0
277277 10.5618i 0.634596i 0.948326 + 0.317298i 0.102776π0.102776\pi
−0.948326 + 0.317298i 0.897224π0.897224\pi
278278 6.82922 15.8390i 0.409589 0.949962i
279279 0 0
280280 9.87787 3.60074i 0.590315 0.215185i
281281 11.5994 0.691961 0.345981 0.938242i 0.387546π-0.387546\pi
0.345981 + 0.938242i 0.387546π0.387546\pi
282282 0 0
283283 21.6234i 1.28538i 0.766128 + 0.642688i 0.222181π0.222181\pi
−0.766128 + 0.642688i 0.777819π0.777819\pi
284284 −16.8737 17.8733i −1.00127 1.06059i
285285 0 0
286286 −4.51595 1.94712i −0.267034 0.115135i
287287 −9.94246 −0.586885
288288 0 0
289289 −6.01240 −0.353671
290290 4.62261 + 1.99310i 0.271449 + 0.117039i
291291 0 0
292292 −18.1834 19.2607i −1.06411 1.12715i
293293 20.1305i 1.17604i 0.808848 + 0.588018i 0.200092π0.200092\pi
−0.808848 + 0.588018i 0.799908π0.799908\pi
294294 0 0
295295 −6.62289 −0.385600
296296 18.5376 6.75742i 1.07747 0.392767i
297297 0 0
298298 −1.70221 + 3.94793i −0.0986062 + 0.228698i
299299 22.4410i 1.29780i
300300 0 0
301301 4.74952i 0.273758i
302302 −2.38521 1.02842i −0.137253 0.0591787i
303303 0 0
304304 28.4806 + 1.64011i 1.63347 + 0.0940668i
305305 16.2529 0.930637
306306 0 0
307307 2.24089i 0.127894i 0.997953 + 0.0639471i 0.0203689π0.0203689\pi
−0.997953 + 0.0639471i 0.979631π0.979631\pi
308308 −2.14863 + 2.02845i −0.122429 + 0.115582i
309309 0 0
310310 0.896143 2.07843i 0.0508975 0.118047i
311311 15.6919 0.889807 0.444904 0.895578i 0.353238π-0.353238\pi
0.444904 + 0.895578i 0.353238π0.353238\pi
312312 0 0
313313 −12.1167 −0.684876 −0.342438 0.939540i 0.611253π-0.611253\pi
−0.342438 + 0.939540i 0.611253π0.611253\pi
314314 −5.63066 + 13.0592i −0.317756 + 0.736973i
315315 0 0
316316 −22.4354 23.7646i −1.26209 1.33686i
317317 11.8958i 0.668132i 0.942550 + 0.334066i 0.108421π0.108421\pi
−0.942550 + 0.334066i 0.891579π0.891579\pi
318318 0 0
319319 −1.41480 −0.0792135
320320 12.9529 + 15.4059i 0.724089 + 0.861218i
321321 0 0
322322 12.3817 + 5.33855i 0.690007 + 0.297506i
323323 23.6407i 1.31540i
324324 0 0
325325 4.62502i 0.256550i
326326 7.21939 16.7440i 0.399845 0.927362i
327327 0 0
328328 −6.51880 17.8830i −0.359941 0.987421i
329329 −1.27403 −0.0702396
330330 0 0
331331 24.7692i 1.36144i −0.732544 0.680719i 0.761667π-0.761667\pi
0.732544 0.680719i 0.238333π-0.238333\pi
332332 20.0751 18.9523i 1.10176 1.04014i
333333 0 0
334334 −5.81058 2.50531i −0.317941 0.137085i
335335 19.2829 1.05354
336336 0 0
337337 −11.2285 −0.611654 −0.305827 0.952087i 0.598933π-0.598933\pi
−0.305827 + 0.952087i 0.598933π0.598933\pi
338338 1.17847 + 0.508115i 0.0641004 + 0.0276378i
339339 0 0
340340 12.1285 11.4501i 0.657760 0.620972i
341341 0.636125i 0.0344481i
342342 0 0
343343 −17.4591 −0.942703
344344 8.54270 3.11403i 0.460591 0.167898i
345345 0 0
346346 0.102359 0.237401i 0.00550285 0.0127628i
347347 16.0230i 0.860160i −0.902791 0.430080i 0.858485π-0.858485\pi
0.902791 0.430080i 0.141515π-0.141515\pi
348348 0 0
349349 9.27513i 0.496486i 0.968698 + 0.248243i 0.0798532π0.0798532\pi
−0.968698 + 0.248243i 0.920147π0.920147\pi
350350 −2.55184 1.10026i −0.136402 0.0588115i
351351 0 0
352352 −5.05722 2.53466i −0.269551 0.135098i
353353 9.58150 0.509972 0.254986 0.966945i 0.417929π-0.417929\pi
0.254986 + 0.966945i 0.417929π0.417929\pi
354354 0 0
355355 30.9211i 1.64112i
356356 1.44132 + 1.52671i 0.0763897 + 0.0809153i
357357 0 0
358358 −11.2498 + 26.0917i −0.594570 + 1.37899i
359359 7.80501 0.411932 0.205966 0.978559i 0.433966π-0.433966\pi
0.205966 + 0.978559i 0.433966π0.433966\pi
360360 0 0
361361 −31.8647 −1.67709
362362 4.27457 9.91402i 0.224666 0.521069i
363363 0 0
364364 −7.47171 + 7.05381i −0.391624 + 0.369720i
365365 33.3212i 1.74411i
366366 0 0
367367 31.4393 1.64112 0.820558 0.571563i 0.193663π-0.193663\pi
0.820558 + 0.571563i 0.193663π0.193663\pi
368368 −1.48405 + 25.7706i −0.0773614 + 1.34338i
369369 0 0
370370 −22.7925 9.82731i −1.18493 0.510897i
371371 19.5782i 1.01645i
372372 0 0
373373 32.0424i 1.65909i −0.558437 0.829547i 0.688599π-0.688599\pi
0.558437 0.829547i 0.311401π-0.311401\pi
374374 −1.85603 + 4.30469i −0.0959729 + 0.222590i
375375 0 0
376376 −0.835322 2.29153i −0.0430785 0.118177i
377377 −4.91987 −0.253386
378378 0 0
379379 11.4465i 0.587965i 0.955811 + 0.293983i 0.0949808π0.0949808\pi
−0.955811 + 0.293983i 0.905019π0.905019\pi
380380 −24.6359 26.0954i −1.26379 1.33867i
381381 0 0
382382 29.8615 + 12.8752i 1.52785 + 0.658752i
383383 6.82520 0.348751 0.174376 0.984679i 0.444209π-0.444209\pi
0.174376 + 0.984679i 0.444209π0.444209\pi
384384 0 0
385385 3.71715 0.189443
386386 0.954865 + 0.411703i 0.0486014 + 0.0209552i
387387 0 0
388388 22.2571 + 23.5756i 1.12993 + 1.19687i
389389 6.65984i 0.337667i 0.985645 + 0.168834i 0.0540000π0.0540000\pi
−0.985645 + 0.168834i 0.946000π0.946000\pi
390390 0 0
391391 21.3912 1.08180
392392 −4.66633 12.8011i −0.235685 0.646553i
393393 0 0
394394 6.25244 14.5013i 0.314993 0.730565i
395395 41.1130i 2.06862i
396396 0 0
397397 2.42189i 0.121551i −0.998151 0.0607757i 0.980643π-0.980643\pi
0.998151 0.0607757i 0.0193574π-0.0193574\pi
398398 −23.5464 10.1524i −1.18028 0.508892i
399399 0 0
400400 0.305858 5.31125i 0.0152929 0.265562i
401401 9.09122 0.453994 0.226997 0.973895i 0.427109π-0.427109\pi
0.226997 + 0.973895i 0.427109π0.427109\pi
402402 0 0
403403 2.21208i 0.110192i
404404 −12.5573 + 11.8550i −0.624748 + 0.589806i
405405 0 0
406406 −1.17040 + 2.71452i −0.0580861 + 0.134719i
407407 6.97588 0.345782
408408 0 0
409409 9.08013 0.448984 0.224492 0.974476i 0.427928π-0.427928\pi
0.224492 + 0.974476i 0.427928π0.427928\pi
410410 −9.48029 + 21.9877i −0.468198 + 1.08589i
411411 0 0
412412 −6.80283 7.20585i −0.335151 0.355007i
413413 3.88914i 0.191372i
414414 0 0
415415 −34.7301 −1.70483
416416 −17.5862 8.81410i −0.862232 0.432147i
417417 0 0
418418 9.26187 + 3.99338i 0.453013 + 0.195323i
419419 2.99512i 0.146321i −0.997320 0.0731607i 0.976691π-0.976691\pi
0.997320 0.0731607i 0.0233086π-0.0233086\pi
420420 0 0
421421 26.6646i 1.29955i 0.760126 + 0.649776i 0.225137π0.225137\pi
−0.760126 + 0.649776i 0.774863π0.774863\pi
422422 −10.3128 + 23.9184i −0.502017 + 1.16433i
423423 0 0
424424 −35.2142 + 12.8365i −1.71015 + 0.623395i
425425 −4.40866 −0.213851
426426 0 0
427427 9.54412i 0.461872i
428428 9.22993 8.71370i 0.446146 0.421193i
429429 0 0
430430 −10.5035 4.52874i −0.506524 0.218395i
431431 −19.5700 −0.942656 −0.471328 0.881958i 0.656225π-0.656225\pi
−0.471328 + 0.881958i 0.656225π0.656225\pi
432432 0 0
433433 −6.49028 −0.311903 −0.155951 0.987765i 0.549844π-0.549844\pi
−0.155951 + 0.987765i 0.549844π0.549844\pi
434434 1.22051 + 0.526239i 0.0585862 + 0.0252603i
435435 0 0
436436 −11.5507 + 10.9047i −0.553179 + 0.522239i
437437 46.0247i 2.20166i
438438 0 0
439439 25.3338 1.20912 0.604559 0.796560i 0.293349π-0.293349\pi
0.604559 + 0.796560i 0.293349π0.293349\pi
440440 2.43716 + 6.68583i 0.116187 + 0.318735i
441441 0 0
442442 −6.45421 + 14.9693i −0.306995 + 0.712016i
443443 1.46196i 0.0694597i 0.999397 + 0.0347299i 0.0110571π0.0110571\pi
−0.999397 + 0.0347299i 0.988943π0.988943\pi
444444 0 0
445445 2.64122i 0.125206i
446446 18.9152 + 8.15556i 0.895662 + 0.386177i
447447 0 0
448448 −9.04677 + 7.60628i −0.427420 + 0.359363i
449449 6.00066 0.283188 0.141594 0.989925i 0.454777π-0.454777\pi
0.141594 + 0.989925i 0.454777π0.454777\pi
450450 0 0
451451 6.72955i 0.316882i
452452 −6.70345 7.10059i −0.315304 0.333983i
453453 0 0
454454 −4.00430 + 9.28719i −0.187931 + 0.435869i
455455 12.9261 0.605986
456456 0 0
457457 17.5689 0.821837 0.410919 0.911672i 0.365208π-0.365208\pi
0.410919 + 0.911672i 0.365208π0.365208\pi
458458 −3.63114 + 8.42172i −0.169672 + 0.393521i
459459 0 0
460460 23.6123 22.2917i 1.10093 1.03936i
461461 26.1231i 1.21668i 0.793678 + 0.608338i 0.208163π0.208163\pi
−0.793678 + 0.608338i 0.791837π0.791837\pi
462462 0 0
463463 20.3258 0.944619 0.472310 0.881433i 0.343420π-0.343420\pi
0.472310 + 0.881433i 0.343420π0.343420\pi
464464 −5.64983 0.325357i −0.262287 0.0151043i
465465 0 0
466466 −14.9066 6.42720i −0.690537 0.297734i
467467 1.29764i 0.0600478i −0.999549 0.0300239i 0.990442π-0.990442\pi
0.999549 0.0300239i 0.00955833π-0.00955833\pi
468468 0 0
469469 11.3234i 0.522868i
470470 −1.21481 + 2.81751i −0.0560349 + 0.129962i
471471 0 0
472472 6.99518 2.54992i 0.321979 0.117370i
473473 3.21471 0.147813
474474 0 0
475475 9.48557i 0.435228i
476476 6.72383 + 7.12217i 0.308186 + 0.326444i
477477 0 0
478478 −0.145495 0.0627320i −0.00665476 0.00286929i
479479 28.5133 1.30281 0.651404 0.758731i 0.274180π-0.274180\pi
0.651404 + 0.758731i 0.274180π0.274180\pi
480480 0 0
481481 24.2582 1.10608
482482 3.81719 + 1.64584i 0.173868 + 0.0749658i
483483 0 0
484484 −1.37296 1.45430i −0.0624072 0.0661045i
485485 40.7861i 1.85200i
486486 0 0
487487 −6.03364 −0.273410 −0.136705 0.990612i 0.543651π-0.543651\pi
−0.136705 + 0.990612i 0.543651π0.543651\pi
488488 −17.1665 + 6.25763i −0.777090 + 0.283270i
489489 0 0
490490 −6.78624 + 15.7394i −0.306571 + 0.711031i
491491 4.15408i 0.187471i −0.995597 0.0937354i 0.970119π-0.970119\pi
0.995597 0.0937354i 0.0298808π-0.0298808\pi
492492 0 0
493493 4.68971i 0.211214i
494494 32.2075 + 13.8867i 1.44909 + 0.624793i
495495 0 0
496496 −0.146288 + 2.54029i −0.00656850 + 0.114062i
497497 −18.1577 −0.814482
498498 0 0
499499 30.1028i 1.34759i −0.738920 0.673793i 0.764664π-0.764664\pi
0.738920 0.673793i 0.235336π-0.235336\pi
500500 13.4283 12.6772i 0.600531 0.566943i
501501 0 0
502502 17.4809 40.5436i 0.780212 1.80955i
503503 −14.0368 −0.625869 −0.312935 0.949775i 0.601312π-0.601312\pi
−0.312935 + 0.949775i 0.601312π0.601312\pi
504504 0 0
505505 21.7242 0.966716
506506 −3.61340 + 8.38057i −0.160635 + 0.372562i
507507 0 0
508508 −5.65683 5.99196i −0.250981 0.265850i
509509 16.9844i 0.752821i −0.926453 0.376411i 0.877158π-0.877158\pi
0.926453 0.376411i 0.122842π-0.122842\pi
510510 0 0
511511 −19.5671 −0.865597
512512 −19.6126 11.2849i −0.866760 0.498725i
513513 0 0
514514 −9.21702 3.97405i −0.406546 0.175288i
515515 12.4662i 0.549326i
516516 0 0
517517 0.862328i 0.0379251i
518518 5.77085 13.3844i 0.253557 0.588075i
519519 0 0
520520 8.47506 + 23.2495i 0.371656 + 1.01956i
521521 3.66607 0.160613 0.0803066 0.996770i 0.474410π-0.474410\pi
0.0803066 + 0.996770i 0.474410π0.474410\pi
522522 0 0
523523 2.91807i 0.127598i −0.997963 0.0637990i 0.979678π-0.979678\pi
0.997963 0.0637990i 0.0203217π-0.0203217\pi
524524 19.3444 18.2624i 0.845063 0.797799i
525525 0 0
526526 −21.2800 9.17516i −0.927851 0.400056i
527527 2.10860 0.0918519
528528 0 0
529529 18.6453 0.810666
530530 43.2970 + 18.6681i 1.88070 + 0.810890i
531531 0 0
532532 15.3239 14.4668i 0.664375 0.627217i
533533 23.4016i 1.01363i
534534 0 0
535535 −15.9679 −0.690352
536536 −20.3668 + 7.42424i −0.879713 + 0.320678i
537537 0 0
538538 9.58947 22.2409i 0.413431 0.958873i
539539 4.81719i 0.207491i
540540 0 0
541541 16.4673i 0.707984i 0.935248 + 0.353992i 0.115176π0.115176\pi
−0.935248 + 0.353992i 0.884824π0.884824\pi
542542 25.3420 + 10.9266i 1.08853 + 0.469337i
543543 0 0
544544 −8.40176 + 16.7635i −0.360223 + 0.718727i
545545 19.9828 0.855971
546546 0 0
547547 25.9536i 1.10969i 0.831952 + 0.554847i 0.187223π0.187223\pi
−0.831952 + 0.554847i 0.812777π0.812777\pi
548548 9.39714 + 9.95386i 0.401426 + 0.425208i
549549 0 0
550550 0.744712 1.72721i 0.0317546 0.0736486i
551551 10.0903 0.429860
552552 0 0
553553 −24.1426 −1.02665
554554 −5.91385 + 13.7160i −0.251255 + 0.582737i
555555 0 0
556556 17.7375 16.7454i 0.752236 0.710163i
557557 13.0152i 0.551473i 0.961233 + 0.275737i 0.0889217π0.0889217\pi
−0.961233 + 0.275737i 0.911078π0.911078\pi
558558 0 0
559559 11.1789 0.472819
560560 14.8440 + 0.854821i 0.627274 + 0.0361228i
561561 0 0
562562 15.0635 + 6.49483i 0.635415 + 0.273968i
563563 27.9837i 1.17937i 0.807632 + 0.589687i 0.200749π0.200749\pi
−0.807632 + 0.589687i 0.799251π0.799251\pi
564564 0 0
565565 12.2841i 0.516795i
566566 −12.1076 + 28.0811i −0.508918 + 1.18034i
567567 0 0
568568 −11.9051 32.6592i −0.499528 1.37035i
569569 10.6048 0.444574 0.222287 0.974981i 0.428648π-0.428648\pi
0.222287 + 0.974981i 0.428648π0.428648\pi
570570 0 0
571571 1.82472i 0.0763622i −0.999271 0.0381811i 0.987844π-0.987844\pi
0.999271 0.0381811i 0.0121564π-0.0121564\pi
572572 −4.77437 5.05722i −0.199627 0.211453i
573573 0 0
574574 −12.9117 5.56707i −0.538925 0.232365i
575575 −8.58298 −0.357935
576576 0 0
577577 23.3377 0.971560 0.485780 0.874081i 0.338536π-0.338536\pi
0.485780 + 0.874081i 0.338536π0.338536\pi
578578 −7.80798 3.36652i −0.324769 0.140029i
579579 0 0
580580 4.88714 + 5.17667i 0.202927 + 0.214949i
581581 20.3944i 0.846103i
582582 0 0
583583 −13.2515 −0.548821
584584 −12.8292 35.1943i −0.530877 1.45635i
585585 0 0
586586 −11.2716 + 26.1424i −0.465627 + 1.07993i
587587 30.3967i 1.25461i 0.778775 + 0.627303i 0.215841π0.215841\pi
−0.778775 + 0.627303i 0.784159π0.784159\pi
588588 0 0
589589 4.53681i 0.186936i
590590 −8.60079 3.70835i −0.354089 0.152670i
591591 0 0
592592 27.8574 + 1.60422i 1.14493 + 0.0659331i
593593 −39.3809 −1.61718 −0.808590 0.588373i 0.799769π-0.799769\pi
−0.808590 + 0.588373i 0.799769π0.799769\pi
594594 0 0
595595 12.3214i 0.505130i
596596 −4.42113 + 4.17385i −0.181096 + 0.170968i
597597 0 0
598598 −12.5654 + 29.1429i −0.513835 + 1.19174i
599599 35.9183 1.46758 0.733792 0.679374i 0.237749π-0.237749\pi
0.733792 + 0.679374i 0.237749π0.237749\pi
600600 0 0
601601 13.8292 0.564104 0.282052 0.959399i 0.408985π-0.408985\pi
0.282052 + 0.959399i 0.408985π0.408985\pi
602602 2.65939 6.16794i 0.108389 0.251386i
603603 0 0
604604 −2.52170 2.67110i −0.102607 0.108685i
605605 2.51595i 0.102288i
606606 0 0
607607 19.7004 0.799615 0.399807 0.916599i 0.369077π-0.369077\pi
0.399807 + 0.916599i 0.369077π0.369077\pi
608608 36.0679 + 18.0770i 1.46274 + 0.733121i
609609 0 0
610610 21.1067 + 9.10046i 0.854586 + 0.368467i
611611 2.99869i 0.121314i
612612 0 0
613613 31.7465i 1.28223i −0.767445 0.641114i 0.778472π-0.778472\pi
0.767445 0.641114i 0.221528π-0.221528\pi
614614 −1.25474 + 2.91012i −0.0506371 + 0.117443i
615615 0 0
616616 −3.92610 + 1.43116i −0.158187 + 0.0576632i
617617 27.5402 1.10873 0.554363 0.832275i 0.312962π-0.312962\pi
0.554363 + 0.832275i 0.312962π0.312962\pi
618618 0 0
619619 19.6108i 0.788224i −0.919063 0.394112i 0.871052π-0.871052\pi
0.919063 0.394112i 0.128948π-0.128948\pi
620620 2.32754 2.19736i 0.0934764 0.0882483i
621621 0 0
622622 20.3782 + 8.78636i 0.817093 + 0.352301i
623623 1.55099 0.0621392
624624 0 0
625625 −29.8811 −1.19525
626626 −15.7353 6.78449i −0.628908 0.271163i
627627 0 0
628628 −14.6244 + 13.8065i −0.583579 + 0.550939i
629629 23.1233i 0.921988i
630630 0 0
631631 17.6518 0.702705 0.351353 0.936243i 0.385722π-0.385722\pi
0.351353 + 0.936243i 0.385722π0.385722\pi
632632 −15.8292 43.4240i −0.629651 1.72731i
633633 0 0
634634 −6.66078 + 15.4484i −0.264533 + 0.613533i
635635 10.3662i 0.411368i
636636 0 0
637637 16.7515i 0.663717i
638638 −1.83732 0.792187i −0.0727403 0.0313630i
639639 0 0
640640 8.19499 + 27.2596i 0.323936 + 1.07753i
641641 −16.7617 −0.662047 −0.331024 0.943622i 0.607394π-0.607394\pi
−0.331024 + 0.943622i 0.607394π0.607394\pi
642642 0 0
643643 22.1367i 0.872984i 0.899708 + 0.436492i 0.143779π0.143779\pi
−0.899708 + 0.436492i 0.856221π0.856221\pi
644644 13.0903 + 13.8658i 0.515829 + 0.546388i
645645 0 0
646646 13.2371 30.7008i 0.520806 1.20791i
647647 −22.7792 −0.895544 −0.447772 0.894148i 0.647782π-0.647782\pi
−0.447772 + 0.894148i 0.647782π0.647782\pi
648648 0 0
649649 2.63236 0.103329
650650 2.58969 6.00627i 0.101576 0.235585i
651651 0 0
652652 18.7509 17.7021i 0.734340 0.693268i
653653 35.9327i 1.40615i −0.711114 0.703077i 0.751809π-0.751809\pi
0.711114 0.703077i 0.248191π-0.248191\pi
654654 0 0
655655 −33.4660 −1.30762
656656 1.54757 26.8737i 0.0604226 1.04924i
657657 0 0
658658 −1.65451 0.713367i −0.0644997 0.0278099i
659659 1.49362i 0.0581831i 0.999577 + 0.0290916i 0.00926144π0.00926144\pi
−0.999577 + 0.0290916i 0.990739π0.990739\pi
660660 0 0
661661 44.3160i 1.72369i −0.507170 0.861846i 0.669308π-0.669308\pi
0.507170 0.861846i 0.330692π-0.330692\pi
662662 13.8690 32.1664i 0.539034 1.25018i
663663 0 0
664664 36.6823 13.3717i 1.42355 0.518921i
665665 −26.5105 −1.02803
666666 0 0
667667 9.13015i 0.353521i
668668 −6.14308 6.50702i −0.237683 0.251764i
669669 0 0
670670 25.0417 + 10.7971i 0.967444 + 0.417127i
671671 −6.45993 −0.249383
672672 0 0
673673 −33.8921 −1.30644 −0.653222 0.757166i 0.726583π-0.726583\pi
−0.653222 + 0.757166i 0.726583π0.726583\pi
674674 −14.5818 6.28715i −0.561671 0.242172i
675675 0 0
676676 1.24591 + 1.31972i 0.0479196 + 0.0507585i
677677 7.57568i 0.291157i −0.989347 0.145578i 0.953496π-0.953496\pi
0.989347 0.145578i 0.0465043π-0.0465043\pi
678678 0 0
679679 23.9507 0.919143
680680 22.1619 8.07858i 0.849870 0.309800i
681681 0 0
682682 −0.356185 + 0.826100i −0.0136390 + 0.0316330i
683683 16.1346i 0.617374i −0.951164 0.308687i 0.900110π-0.900110\pi
0.951164 0.308687i 0.0998896π-0.0998896\pi
684684 0 0
685685 17.2203i 0.657953i
686686 −22.6732 9.77586i −0.865666 0.373244i
687687 0 0
688688 12.8376 + 0.739276i 0.489428 + 0.0281847i
689689 −46.0812 −1.75555
690690 0 0
691691 10.5531i 0.401457i −0.979647 0.200729i 0.935669π-0.935669\pi
0.979647 0.200729i 0.0643309π-0.0643309\pi
692692 0.265856 0.250987i 0.0101063 0.00954108i
693693 0 0
694694 8.97175 20.8082i 0.340563 0.789869i
695695 −30.6860 −1.16399
696696 0 0
697697 −22.3068 −0.844931
698698 −5.19341 + 12.0451i −0.196574 + 0.455914i
699699 0 0
700700 −2.69787 2.85770i −0.101970 0.108011i
701701 43.8782i 1.65726i 0.559800 + 0.828628i 0.310878π0.310878\pi
−0.559800 + 0.828628i 0.689122π0.689122\pi
702702 0 0
703703 −49.7516 −1.87642
704704 −5.14831 6.12331i −0.194034 0.230781i
705705 0 0
706706 12.4430 + 5.36496i 0.468297 + 0.201913i
707707 12.7570i 0.479778i
708708 0 0
709709 4.31359i 0.162000i −0.996714 0.0810002i 0.974189π-0.974189\pi
0.996714 0.0810002i 0.0258115π-0.0258115\pi
710710 −17.3136 + 40.1555i −0.649768 + 1.50701i
711711 0 0
712712 1.01691 + 2.78969i 0.0381104 + 0.104548i
713713 4.10511 0.153738
714714 0 0
715715 8.74905i 0.327196i
716716 −29.2190 + 27.5848i −1.09196 + 1.03089i
717717 0 0
718718 10.1359 + 4.37025i 0.378270 + 0.163096i
719719 0.793243 0.0295830 0.0147915 0.999891i 0.495292π-0.495292\pi
0.0147915 + 0.999891i 0.495292π0.495292\pi
720720 0 0
721721 −7.32048 −0.272629
722722 −41.3809 17.8420i −1.54004 0.664009i
723723 0 0
724724 11.1023 10.4813i 0.412614 0.389536i
725725 1.88170i 0.0698845i
726726 0 0
727727 −38.5654 −1.43031 −0.715155 0.698966i 0.753644π-0.753644\pi
−0.715155 + 0.698966i 0.753644π0.753644\pi
728728 −13.6527 + 4.97678i −0.506004 + 0.184452i
729729 0 0
730730 −18.6575 + 43.2724i −0.690546 + 1.60158i
731731 10.6560i 0.394125i
732732 0 0
733733 35.3659i 1.30627i −0.757242 0.653134i 0.773454π-0.773454\pi
0.757242 0.653134i 0.226546π-0.226546\pi
734734 40.8284 + 17.6038i 1.50701 + 0.649766i
735735 0 0
736736 −16.3569 + 32.6359i −0.602925 + 1.20297i
737737 −7.66426 −0.282317
738738 0 0
739739 41.6624i 1.53258i 0.642497 + 0.766288i 0.277899π0.277899\pi
−0.642497 + 0.766288i 0.722101π0.722101\pi
740740 −24.0968 25.5244i −0.885816 0.938294i
741741 0 0
742742 −10.9624 + 25.4251i −0.402442 + 0.933385i
743743 −33.2692 −1.22053 −0.610264 0.792198i 0.708937π-0.708937\pi
−0.610264 + 0.792198i 0.708937π0.708937\pi
744744 0 0
745745 7.64859 0.280223
746746 17.9415 41.6118i 0.656885 1.52351i
747747 0 0
748748 −4.82064 + 4.55102i −0.176260 + 0.166402i
749749 9.37676i 0.342619i
750750 0 0
751751 −3.69284 −0.134754 −0.0673768 0.997728i 0.521463π-0.521463\pi
−0.0673768 + 0.997728i 0.521463π0.521463\pi
752752 0.198307 3.44361i 0.00723150 0.125575i
753753 0 0
754754 −6.38916 2.75478i −0.232680 0.100323i
755755 4.62103i 0.168176i
756756 0 0
757757 19.2210i 0.698600i −0.937011 0.349300i 0.886419π-0.886419\pi
0.937011 0.349300i 0.113581π-0.113581\pi
758758 −6.40920 + 14.8649i −0.232793 + 0.539917i
759759 0 0
760760 −17.3817 47.6830i −0.630500 1.72964i
761761 9.79341 0.355011 0.177505 0.984120i 0.443197π-0.443197\pi
0.177505 + 0.984120i 0.443197π0.443197\pi
762762 0 0
763763 11.7345i 0.424816i
764764 31.5703 + 33.4406i 1.14217 + 1.20984i
765765 0 0
766766 8.86351 + 3.82163i 0.320252 + 0.138081i
767767 9.15386 0.330527
768768 0 0
769769 −48.2853 −1.74121 −0.870606 0.491981i 0.836273π-0.836273\pi
−0.870606 + 0.491981i 0.836273π0.836273\pi
770770 4.82726 + 2.08134i 0.173962 + 0.0750063i
771771 0 0
772772 1.00951 + 1.06931i 0.0363329 + 0.0384854i
773773 21.0475i 0.757025i −0.925596 0.378513i 0.876436π-0.876436\pi
0.925596 0.378513i 0.123564π-0.123564\pi
774774 0 0
775775 −0.846053 −0.0303911
776776 15.7033 + 43.0788i 0.563717 + 1.54644i
777777 0 0
778778 −3.72904 + 8.64877i −0.133692 + 0.310073i
779779 47.9948i 1.71959i
780780 0 0
781781 12.2900i 0.439771i
782782 27.7795 + 11.9775i 0.993394 + 0.428316i
783783 0 0
784784 1.10779 19.2369i 0.0395641 0.687032i
785785 25.3005 0.903012
786786 0 0
787787 18.2077i 0.649034i −0.945880 0.324517i 0.894798π-0.894798\pi
0.945880 0.324517i 0.105202π-0.105202\pi
788788 16.2394 15.3311i 0.578504 0.546149i
789789 0 0
790790 −23.0204 + 53.3912i −0.819027 + 1.89957i
791791 −7.21354 −0.256484
792792 0 0
793793 −22.4640 −0.797719
794794 1.35609 3.14518i 0.0481258 0.111618i
795795 0 0
796796 −24.8939 26.3687i −0.882339 0.934612i
797797 40.2274i 1.42493i 0.701709 + 0.712463i 0.252421π0.252421\pi
−0.701709 + 0.712463i 0.747579π0.747579\pi
798798 0 0
799799 −2.85840 −0.101123
800800 3.37112 6.72617i 0.119187 0.237806i
801801 0 0
802802 11.8063 + 5.09044i 0.416894 + 0.179750i
803803 13.2440i 0.467370i
804804 0 0
805805 23.9879i 0.845464i
806806 −1.23861 + 2.87271i −0.0436281 + 0.101187i
807807 0 0
808808 −22.9454 + 8.36419i −0.807216 + 0.294251i
809809 −27.8044 −0.977552 −0.488776 0.872409i 0.662556π-0.662556\pi
−0.488776 + 0.872409i 0.662556π0.662556\pi
810810 0 0
811811 7.59987i 0.266868i −0.991058 0.133434i 0.957400π-0.957400\pi
0.991058 0.133434i 0.0426004π-0.0426004\pi
812812 −3.03988 + 2.86985i −0.106679 + 0.100712i
813813 0 0
814814 9.05920 + 3.90600i 0.317525 + 0.136905i
815815 −32.4392 −1.13629
816816 0 0
817817 −22.9271 −0.802119
818818 11.7919 + 5.08423i 0.412293 + 0.177766i
819819 0 0
820820 −24.6231 + 23.2459i −0.859874 + 0.811781i
821821 26.6456i 0.929937i −0.885327 0.464968i 0.846066π-0.846066\pi
0.885327 0.464968i 0.153934π-0.153934\pi
822822 0 0
823823 35.5588 1.23950 0.619751 0.784799i 0.287234π-0.287234\pi
0.619751 + 0.784799i 0.287234π0.287234\pi
824824 −4.79969 13.1669i −0.167205 0.458692i
825825 0 0
826826 2.17764 5.05061i 0.0757698 0.175733i
827827 3.03188i 0.105429i 0.998610 + 0.0527145i 0.0167873π0.0167873\pi
−0.998610 + 0.0527145i 0.983213π0.983213\pi
828828 0 0
829829 35.0569i 1.21758i 0.793332 + 0.608789i 0.208344π0.208344\pi
−0.793332 + 0.608789i 0.791656π0.791656\pi
830830 −45.1021 19.4464i −1.56552 0.674994i
831831 0 0
832832 −17.9029 21.2934i −0.620672 0.738215i
833833 −15.9678 −0.553252
834834 0 0
835835 11.2572i 0.389572i
836836 9.79187 + 10.3720i 0.338659 + 0.358722i
837837 0 0
838838 1.67706 3.88960i 0.0579329 0.134364i
839839 6.41520 0.221477 0.110739 0.993850i 0.464678π-0.464678\pi
0.110739 + 0.993850i 0.464678π0.464678\pi
840840 0 0
841841 26.9983 0.930977
842842 −14.9303 + 34.6278i −0.514531 + 1.19335i
843843 0 0
844844 −26.7852 + 25.2871i −0.921985 + 0.870419i
845845 2.28313i 0.0785421i
846846 0 0
847847 −1.47743 −0.0507652
848848 −52.9183 3.04740i −1.81722 0.104648i
849849 0 0
850850 −5.72529 2.46854i −0.196376 0.0846701i
851851 45.0176i 1.54318i
852852 0 0
853853 10.5605i 0.361585i 0.983521 + 0.180793i 0.0578663π0.0578663\pi
−0.983521 + 0.180793i 0.942134π0.942134\pi
854854 −5.34403 + 12.3944i −0.182869 + 0.424128i
855855 0 0
856856 16.8655 6.14790i 0.576450 0.210131i
857857 5.03900 0.172129 0.0860644 0.996290i 0.472571π-0.472571\pi
0.0860644 + 0.996290i 0.472571π0.472571\pi
858858 0 0
859859 24.5174i 0.836521i 0.908327 + 0.418261i 0.137360π0.137360\pi
−0.908327 + 0.418261i 0.862640π0.862640\pi
860860 −11.1046 11.7624i −0.378663 0.401096i
861861 0 0
862862 −25.4145 10.9578i −0.865623 0.373225i
863863 −17.8155 −0.606448 −0.303224 0.952919i 0.598063π-0.598063\pi
−0.303224 + 0.952919i 0.598063π0.598063\pi
864864 0 0
865865 −0.459934 −0.0156382
866866 −8.42857 3.63409i −0.286414 0.123492i
867867 0 0
868868 1.29035 + 1.36679i 0.0437973 + 0.0463920i
869869 16.3409i 0.554328i
870870 0 0
871871 −26.6520 −0.903067
872872 −21.1061 + 7.69373i −0.714743 + 0.260542i
873873 0 0
874874 25.7706 59.7698i 0.871703 2.02174i
875875 13.6419i 0.461180i
876876 0 0
877877 22.4071i 0.756634i 0.925676 + 0.378317i 0.123497π0.123497\pi
−0.925676 + 0.378317i 0.876503π0.876503\pi
878878 32.8997 + 14.1852i 1.11031 + 0.478726i
879879 0 0
880880 −0.578585 + 10.0472i −0.0195041 + 0.338690i
881881 12.0016 0.404345 0.202172 0.979350i 0.435200π-0.435200\pi
0.202172 + 0.979350i 0.435200π0.435200\pi
882882 0 0
883883 27.9363i 0.940130i −0.882632 0.470065i 0.844230π-0.844230\pi
0.882632 0.470065i 0.155770π-0.155770\pi
884884 −16.7635 + 15.8259i −0.563816 + 0.532282i
885885 0 0
886886 −0.818593 + 1.89857i −0.0275012 + 0.0637836i
887887 48.6903 1.63486 0.817430 0.576028i 0.195398π-0.195398\pi
0.817430 + 0.576028i 0.195398π0.195398\pi
888888 0 0
889889 −6.08728 −0.204161
890890 1.47890 3.43001i 0.0495727 0.114974i
891891 0 0
892892 19.9976 + 21.1824i 0.669570 + 0.709238i
893893 6.15008i 0.205804i
894894 0 0
895895 50.5492 1.68967
896896 −16.0075 + 4.81232i −0.534774 + 0.160768i
897897 0 0
898898 7.79272 + 3.35994i 0.260047 + 0.112123i
899899 0.899988i 0.0300163i
900900 0 0
901901 43.9254i 1.46337i
902902 3.76807 8.73930i 0.125463 0.290987i
903903 0 0
904904 −4.72958 12.9746i −0.157303 0.431529i
905905 −19.2071 −0.638465
906906 0 0
907907 48.2773i 1.60302i −0.597981 0.801510i 0.704030π-0.704030\pi
0.597981 0.801510i 0.295970π-0.295970\pi
908908 −10.4003 + 9.81864i −0.345147 + 0.325843i
909909 0 0
910910 16.7865 + 7.23772i 0.556466 + 0.239928i
911911 −43.1688 −1.43025 −0.715123 0.698998i 0.753630π-0.753630\pi
−0.715123 + 0.698998i 0.753630π0.753630\pi
912912 0 0
913913 13.8040 0.456844
914914 22.8157 + 9.83732i 0.754677 + 0.325390i
915915 0 0
916916 −9.43113 + 8.90364i −0.311613 + 0.294185i
917917 19.6521i 0.648970i
918918 0 0
919919 36.1457 1.19234 0.596168 0.802860i 0.296689π-0.296689\pi
0.596168 + 0.802860i 0.296689π0.296689\pi
920920 43.1458 15.7278i 1.42247 0.518529i
921921 0 0
922922 −14.6271 + 33.9247i −0.481718 + 1.11725i
923923 42.7377i 1.40673i
924924 0 0
925925 9.27801i 0.305059i
926926 26.3960 + 11.3810i 0.867426 + 0.374003i
927927 0 0
928928 −7.15495 3.58603i −0.234873 0.117717i
929929 −26.1901 −0.859269 −0.429634 0.903003i 0.641358π-0.641358\pi
−0.429634 + 0.903003i 0.641358π0.641358\pi
930930 0 0
931931 34.3560i 1.12597i
932932 −15.7597 16.6933i −0.516225 0.546808i
933933 0 0
934934 0.726588 1.68518i 0.0237747 0.0551407i
935935 8.33976 0.272739
936936 0 0
937937 −14.6547 −0.478749 −0.239375 0.970927i 0.576942π-0.576942\pi
−0.239375 + 0.970927i 0.576942π0.576942\pi
938938 −6.34032 + 14.7051i −0.207019 + 0.480139i
939939 0 0
940940 −3.15521 + 2.97874i −0.102912 + 0.0971557i
941941 5.50159i 0.179347i −0.995971 0.0896734i 0.971418π-0.971418\pi
0.995971 0.0896734i 0.0285823π-0.0285823\pi
942942 0 0
943943 −43.4279 −1.41421
944944 10.5120 + 0.605356i 0.342137 + 0.0197026i
945945 0 0
946946 4.17477 + 1.80001i 0.135733 + 0.0585233i
947947 1.00019i 0.0325019i 0.999868 + 0.0162509i 0.00517306π0.00517306\pi
−0.999868 + 0.0162509i 0.994827π0.994827\pi
948948 0 0
949949 46.0551i 1.49501i
950950 −5.31125 + 12.3184i −0.172320 + 0.399661i
951951 0 0
952952 4.74396 + 13.0140i 0.153753 + 0.421788i
953953 −55.6770 −1.80356 −0.901778 0.432200i 0.857737π-0.857737\pi
−0.901778 + 0.432200i 0.857737π0.857737\pi
954954 0 0
955955 57.8526i 1.87207i
956956 −0.153820 0.162933i −0.00497490 0.00526964i
957957 0 0
958958 37.0287 + 15.9654i 1.19634 + 0.515820i
959959 10.1122 0.326540
960960 0 0
961961 −30.5953 −0.986947
962962 31.5028 + 13.5829i 1.01569 + 0.437929i
963963 0 0
964964 4.03563 + 4.27471i 0.129979 + 0.137679i
965965 1.84992i 0.0595511i
966966 0 0
967967 −25.9869 −0.835682 −0.417841 0.908520i 0.637213π-0.637213\pi
−0.417841 + 0.908520i 0.637213π0.637213\pi
968968 −0.968683 2.65738i −0.0311347 0.0854113i
969969 0 0
970970 22.8373 52.9667i 0.733262 1.70066i
971971 16.0587i 0.515349i −0.966232 0.257675i 0.917044π-0.917044\pi
0.966232 0.257675i 0.0829563π-0.0829563\pi
972972 0 0
973973 18.0196i 0.577683i
974974 −7.83556 3.37841i −0.251068 0.108251i
975975 0 0
976976 −25.7970 1.48557i −0.825742 0.0475519i
977977 5.40488 0.172918 0.0864588 0.996255i 0.472445π-0.472445\pi
0.0864588 + 0.996255i 0.472445π0.472445\pi
978978 0 0
979979 1.04979i 0.0335514i
980980 −17.6258 + 16.6400i −0.563037 + 0.531546i
981981 0 0
982982 2.32599 5.39467i 0.0742252 0.172151i
983983 36.4792 1.16350 0.581752 0.813366i 0.302367π-0.302367\pi
0.581752 + 0.813366i 0.302367π0.302367\pi
984984 0 0
985985 −28.0943 −0.895159
986986 −2.62590 + 6.09027i −0.0836258 + 0.193954i
987987 0 0
988988 34.0506 + 36.0679i 1.08329 + 1.14747i
989989 20.7455i 0.659670i
990990 0 0
991991 48.2714 1.53339 0.766696 0.642010i 0.221899π-0.221899\pi
0.766696 + 0.642010i 0.221899π0.221899\pi
992992 −1.61236 + 3.21702i −0.0511924 + 0.102141i
993993 0 0
994994 −23.5804 10.1670i −0.747924 0.322478i
995995 45.6181i 1.44619i
996996 0 0
997997 8.78994i 0.278380i −0.990266 0.139190i 0.955550π-0.955550\pi
0.990266 0.139190i 0.0444499π-0.0444499\pi
998998 16.8554 39.0929i 0.533549 1.23746i
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 792.2.f.g.397.10 10
3.2 odd 2 88.2.c.a.45.1 10
4.3 odd 2 3168.2.f.g.1585.2 10
8.3 odd 2 3168.2.f.g.1585.9 10
8.5 even 2 inner 792.2.f.g.397.9 10
12.11 even 2 352.2.c.a.177.5 10
24.5 odd 2 88.2.c.a.45.2 yes 10
24.11 even 2 352.2.c.a.177.6 10
33.2 even 10 968.2.o.h.565.4 40
33.5 odd 10 968.2.o.g.245.10 40
33.8 even 10 968.2.o.h.493.8 40
33.14 odd 10 968.2.o.g.493.3 40
33.17 even 10 968.2.o.h.245.1 40
33.20 odd 10 968.2.o.g.565.7 40
33.26 odd 10 968.2.o.g.269.6 40
33.29 even 10 968.2.o.h.269.5 40
33.32 even 2 968.2.c.d.485.10 10
48.5 odd 4 2816.2.a.r.1.3 5
48.11 even 4 2816.2.a.q.1.3 5
48.29 odd 4 2816.2.a.o.1.3 5
48.35 even 4 2816.2.a.p.1.3 5
132.131 odd 2 3872.2.c.f.1937.5 10
264.5 odd 10 968.2.o.g.245.7 40
264.29 even 10 968.2.o.h.269.8 40
264.53 odd 10 968.2.o.g.565.10 40
264.101 even 10 968.2.o.h.565.1 40
264.125 odd 10 968.2.o.g.269.3 40
264.131 odd 2 3872.2.c.f.1937.6 10
264.149 even 10 968.2.o.h.245.4 40
264.173 even 10 968.2.o.h.493.5 40
264.197 even 2 968.2.c.d.485.9 10
264.245 odd 10 968.2.o.g.493.6 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.c.a.45.1 10 3.2 odd 2
88.2.c.a.45.2 yes 10 24.5 odd 2
352.2.c.a.177.5 10 12.11 even 2
352.2.c.a.177.6 10 24.11 even 2
792.2.f.g.397.9 10 8.5 even 2 inner
792.2.f.g.397.10 10 1.1 even 1 trivial
968.2.c.d.485.9 10 264.197 even 2
968.2.c.d.485.10 10 33.32 even 2
968.2.o.g.245.7 40 264.5 odd 10
968.2.o.g.245.10 40 33.5 odd 10
968.2.o.g.269.3 40 264.125 odd 10
968.2.o.g.269.6 40 33.26 odd 10
968.2.o.g.493.3 40 33.14 odd 10
968.2.o.g.493.6 40 264.245 odd 10
968.2.o.g.565.7 40 33.20 odd 10
968.2.o.g.565.10 40 264.53 odd 10
968.2.o.h.245.1 40 33.17 even 10
968.2.o.h.245.4 40 264.149 even 10
968.2.o.h.269.5 40 33.29 even 10
968.2.o.h.269.8 40 264.29 even 10
968.2.o.h.493.5 40 264.173 even 10
968.2.o.h.493.8 40 33.8 even 10
968.2.o.h.565.1 40 264.101 even 10
968.2.o.h.565.4 40 33.2 even 10
2816.2.a.o.1.3 5 48.29 odd 4
2816.2.a.p.1.3 5 48.35 even 4
2816.2.a.q.1.3 5 48.11 even 4
2816.2.a.r.1.3 5 48.5 odd 4
3168.2.f.g.1585.2 10 4.3 odd 2
3168.2.f.g.1585.9 10 8.3 odd 2
3872.2.c.f.1937.5 10 132.131 odd 2
3872.2.c.f.1937.6 10 264.131 odd 2