Properties

Label 115.5.c.c.114.12
Level 115115
Weight 55
Character 115.114
Analytic conductor 11.88811.888
Analytic rank 00
Dimension 4444
Inner twists 44

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,5,Mod(114,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: N N == 115=523 115 = 5 \cdot 23
Weight: k k == 5 5
Character orbit: [χ][\chi] == 115.c (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: 11.887545754611.8875457546
Analytic rank: 00
Dimension: 4444
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

Embedding invariants

Embedding label 114.12
Character χ\chi == 115.114
Dual form 115.5.c.c.114.34

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) == q4.94563iq26.82837iq38.45927q4+(22.1235+11.6426i)q533.7706q6+47.4193q737.2937iq8+34.3734q9+(57.5800109.415i)q10121.460iq11+57.7631iq1225.2927iq13234.518iq14+(79.4999151.068i)q15319.789q16+130.920q17169.998iq18+292.655iq19+(187.14998.4878i)q20323.797iq21600.696q22+(215.187+483.255i)q23254.655q24+(353.900+515.150i)q25125.088q26787.812iq27401.133q28790.956q29+(747.125393.177i)q30+365.318q31+984.861iq32829.373q33647.481iq34+(1049.08+552.083i)q35290.774q36471.550q37+1447.36q38172.708q39+(434.195825.067i)q401821.57q411601.38q427.12217q43+1027.46iq44+(760.460+400.195i)q45+(2390.00+1064.24i)q462540.24iq47+2183.64iq48152.410q49+(2547.741750.26i)q50893.969iq51+213.958iq52+2852.07q533896.23q54+(1414.112687.12i)q551768.44iq56+1998.36q57+3911.78iq584053.87q59+(672.512+1277.92i)q60+3186.31iq611806.73iq62+1629.96q63245.867q64+(294.472559.563i)q65+4101.77iq66+6094.08q671107.49q68+(3299.84+1469.38i)q69+(2730.405188.37i)q70+4387.94q711281.91iq72+3217.13iq73+2332.11iq74+(3517.642416.56i)q752475.65iq765759.54iq77+854.149iq78+8824.58iq79+(7074.863723.17i)q802595.23q81+9008.82iq82+12862.0q83+2739.08iq84+(2896.41+1524.24i)q85+35.2236iq86+5400.94iq874529.68q88+5490.66iq89+(1979.223760.95i)q901199.36iq91+(1820.334087.99i)q922494.53iq9312563.1q94+(3407.26+6474.56i)q95+6724.99q9615520.6q97+753.765iq984174.98iq99+O(q100)q-4.94563i q^{2} -6.82837i q^{3} -8.45927 q^{4} +(22.1235 + 11.6426i) q^{5} -33.7706 q^{6} +47.4193 q^{7} -37.2937i q^{8} +34.3734 q^{9} +(57.5800 - 109.415i) q^{10} -121.460i q^{11} +57.7631i q^{12} -25.2927i q^{13} -234.518i q^{14} +(79.4999 - 151.068i) q^{15} -319.789 q^{16} +130.920 q^{17} -169.998i q^{18} +292.655i q^{19} +(-187.149 - 98.4878i) q^{20} -323.797i q^{21} -600.696 q^{22} +(-215.187 + 483.255i) q^{23} -254.655 q^{24} +(353.900 + 515.150i) q^{25} -125.088 q^{26} -787.812i q^{27} -401.133 q^{28} -790.956 q^{29} +(-747.125 - 393.177i) q^{30} +365.318 q^{31} +984.861i q^{32} -829.373 q^{33} -647.481i q^{34} +(1049.08 + 552.083i) q^{35} -290.774 q^{36} -471.550 q^{37} +1447.36 q^{38} -172.708 q^{39} +(434.195 - 825.067i) q^{40} -1821.57 q^{41} -1601.38 q^{42} -7.12217 q^{43} +1027.46i q^{44} +(760.460 + 400.195i) q^{45} +(2390.00 + 1064.24i) q^{46} -2540.24i q^{47} +2183.64i q^{48} -152.410 q^{49} +(2547.74 - 1750.26i) q^{50} -893.969i q^{51} +213.958i q^{52} +2852.07 q^{53} -3896.23 q^{54} +(1414.11 - 2687.12i) q^{55} -1768.44i q^{56} +1998.36 q^{57} +3911.78i q^{58} -4053.87 q^{59} +(-672.512 + 1277.92i) q^{60} +3186.31i q^{61} -1806.73i q^{62} +1629.96 q^{63} -245.867 q^{64} +(294.472 - 559.563i) q^{65} +4101.77i q^{66} +6094.08 q^{67} -1107.49 q^{68} +(3299.84 + 1469.38i) q^{69} +(2730.40 - 5188.37i) q^{70} +4387.94 q^{71} -1281.91i q^{72} +3217.13i q^{73} +2332.11i q^{74} +(3517.64 - 2416.56i) q^{75} -2475.65i q^{76} -5759.54i q^{77} +854.149i q^{78} +8824.58i q^{79} +(-7074.86 - 3723.17i) q^{80} -2595.23 q^{81} +9008.82i q^{82} +12862.0 q^{83} +2739.08i q^{84} +(2896.41 + 1524.24i) q^{85} +35.2236i q^{86} +5400.94i q^{87} -4529.68 q^{88} +5490.66i q^{89} +(1979.22 - 3760.95i) q^{90} -1199.36i q^{91} +(1820.33 - 4087.99i) q^{92} -2494.53i q^{93} -12563.1 q^{94} +(-3407.26 + 6474.56i) q^{95} +6724.99 q^{96} -15520.6 q^{97} +753.765i q^{98} -4174.98i q^{99} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 44q360q4+160q61128q9+2136q163940q242540q25+3144q26348q294480q311440q353728q361800q39+7176q41+5372q46+9580q49+39960q96+O(q100) 44 q - 360 q^{4} + 160 q^{6} - 1128 q^{9} + 2136 q^{16} - 3940 q^{24} - 2540 q^{25} + 3144 q^{26} - 348 q^{29} - 4480 q^{31} - 1440 q^{35} - 3728 q^{36} - 1800 q^{39} + 7176 q^{41} + 5372 q^{46} + 9580 q^{49}+ \cdots - 39960 q^{96}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/115Z)×\left(\mathbb{Z}/115\mathbb{Z}\right)^\times.

nn 4747 5151
χ(n)\chi(n) 1-1 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 4.94563i 1.23641i −0.786018 0.618204i 0.787861π-0.787861\pi
0.786018 0.618204i 0.212139π-0.212139\pi
33 6.82837i 0.758708i −0.925252 0.379354i 0.876146π-0.876146\pi
0.925252 0.379354i 0.123854π-0.123854\pi
44 −8.45927 −0.528705
55 22.1235 + 11.6426i 0.884941 + 0.465703i
66 −33.7706 −0.938072
77 47.4193 0.967741 0.483870 0.875140i 0.339231π-0.339231\pi
0.483870 + 0.875140i 0.339231π0.339231\pi
88 37.2937i 0.582713i
99 34.3734 0.424362
1010 57.5800 109.415i 0.575800 1.09415i
1111 121.460i 1.00380i −0.864926 0.501900i 0.832634π-0.832634\pi
0.864926 0.501900i 0.167366π-0.167366\pi
1212 57.7631i 0.401132i
1313 25.2927i 0.149661i −0.997196 0.0748304i 0.976158π-0.976158\pi
0.997196 0.0748304i 0.0238416π-0.0238416\pi
1414 234.518i 1.19652i
1515 79.4999 151.068i 0.353333 0.671412i
1616 −319.789 −1.24918
1717 130.920 0.453010 0.226505 0.974010i 0.427270π-0.427270\pi
0.226505 + 0.974010i 0.427270π0.427270\pi
1818 169.998i 0.524685i
1919 292.655i 0.810679i 0.914166 + 0.405339i 0.132847π0.132847\pi
−0.914166 + 0.405339i 0.867153π0.867153\pi
2020 −187.149 98.4878i −0.467872 0.246220i
2121 323.797i 0.734233i
2222 −600.696 −1.24111
2323 −215.187 + 483.255i −0.406781 + 0.913526i
2424 −254.655 −0.442109
2525 353.900 + 515.150i 0.566241 + 0.824240i
2626 −125.088 −0.185042
2727 787.812i 1.08067i
2828 −401.133 −0.511649
2929 −790.956 −0.940495 −0.470247 0.882535i 0.655835π-0.655835\pi
−0.470247 + 0.882535i 0.655835π0.655835\pi
3030 −747.125 393.177i −0.830139 0.436864i
3131 365.318 0.380144 0.190072 0.981770i 0.439128π-0.439128\pi
0.190072 + 0.981770i 0.439128π0.439128\pi
3232 984.861i 0.961778i
3333 −829.373 −0.761591
3434 647.481i 0.560105i
3535 1049.08 + 552.083i 0.856393 + 0.450680i
3636 −290.774 −0.224362
3737 −471.550 −0.344449 −0.172224 0.985058i 0.555095π-0.555095\pi
−0.172224 + 0.985058i 0.555095π0.555095\pi
3838 1447.36 1.00233
3939 −172.708 −0.113549
4040 434.195 825.067i 0.271372 0.515667i
4141 −1821.57 −1.08362 −0.541812 0.840500i 0.682261π-0.682261\pi
−0.541812 + 0.840500i 0.682261π0.682261\pi
4242 −1601.38 −0.907811
4343 −7.12217 −0.00385190 −0.00192595 0.999998i 0.500613π-0.500613\pi
−0.00192595 + 0.999998i 0.500613π0.500613\pi
4444 1027.46i 0.530714i
4545 760.460 + 400.195i 0.375536 + 0.197627i
4646 2390.00 + 1064.24i 1.12949 + 0.502948i
4747 2540.24i 1.14995i −0.818171 0.574975i 0.805012π-0.805012\pi
0.818171 0.574975i 0.194988π-0.194988\pi
4848 2183.64i 0.947760i
4949 −152.410 −0.0634778
5050 2547.74 1750.26i 1.01910 0.700104i
5151 893.969i 0.343702i
5252 213.958i 0.0791264i
5353 2852.07 1.01533 0.507666 0.861554i 0.330509π-0.330509\pi
0.507666 + 0.861554i 0.330509π0.330509\pi
5454 −3896.23 −1.33616
5555 1414.11 2687.12i 0.467473 0.888304i
5656 1768.44i 0.563915i
5757 1998.36 0.615068
5858 3911.78i 1.16284i
5959 −4053.87 −1.16457 −0.582285 0.812985i 0.697841π-0.697841\pi
−0.582285 + 0.812985i 0.697841π0.697841\pi
6060 −672.512 + 1277.92i −0.186809 + 0.354978i
6161 3186.31i 0.856304i 0.903707 + 0.428152i 0.140835π0.140835\pi
−0.903707 + 0.428152i 0.859165π0.859165\pi
6262 1806.73i 0.470013i
6363 1629.96 0.410673
6464 −245.867 −0.0600262
6565 294.472 559.563i 0.0696976 0.132441i
6666 4101.77i 0.941638i
6767 6094.08 1.35756 0.678779 0.734342i 0.262509π-0.262509\pi
0.678779 + 0.734342i 0.262509π0.262509\pi
6868 −1107.49 −0.239508
6969 3299.84 + 1469.38i 0.693099 + 0.308628i
7070 2730.40 5188.37i 0.557225 1.05885i
7171 4387.94 0.870449 0.435225 0.900322i 0.356669π-0.356669\pi
0.435225 + 0.900322i 0.356669π0.356669\pi
7272 1281.91i 0.247282i
7373 3217.13i 0.603702i 0.953355 + 0.301851i 0.0976044π0.0976044\pi
−0.953355 + 0.301851i 0.902396π0.902396\pi
7474 2332.11i 0.425879i
7575 3517.64 2416.56i 0.625357 0.429611i
7676 2475.65i 0.428610i
7777 5759.54i 0.971419i
7878 854.149i 0.140393i
7979 8824.58i 1.41397i 0.707229 + 0.706984i 0.249945π0.249945\pi
−0.707229 + 0.706984i 0.750055π0.750055\pi
8080 −7074.86 3723.17i −1.10545 0.581746i
8181 −2595.23 −0.395554
8282 9008.82i 1.33980i
8383 12862.0 1.86704 0.933519 0.358527i 0.116721π-0.116721\pi
0.933519 + 0.358527i 0.116721π0.116721\pi
8484 2739.08i 0.388192i
8585 2896.41 + 1524.24i 0.400887 + 0.210968i
8686 35.2236i 0.00476252i
8787 5400.94i 0.713561i
8888 −4529.68 −0.584928
8989 5490.66i 0.693178i 0.938017 + 0.346589i 0.112660π0.112660\pi
−0.938017 + 0.346589i 0.887340π0.887340\pi
9090 1979.22 3760.95i 0.244348 0.464315i
9191 1199.36i 0.144833i
9292 1820.33 4087.99i 0.215067 0.482985i
9393 2494.53i 0.288418i
9494 −12563.1 −1.42181
9595 −3407.26 + 6474.56i −0.377536 + 0.717403i
9696 6724.99 0.729708
9797 −15520.6 −1.64955 −0.824775 0.565461i 0.808698π-0.808698\pi
−0.824775 + 0.565461i 0.808698π0.808698\pi
9898 753.765i 0.0784845i
9999 4174.98i 0.425975i
100100 −2993.74 4357.80i −0.299374 0.435780i
101101 3046.74 0.298671 0.149336 0.988787i 0.452287π-0.452287\pi
0.149336 + 0.988787i 0.452287π0.452287\pi
102102 −4421.24 −0.424956
103103 −6210.09 −0.585361 −0.292680 0.956210i 0.594547π-0.594547\pi
−0.292680 + 0.956210i 0.594547π0.594547\pi
104104 −943.257 −0.0872094
105105 3769.83 7163.52i 0.341935 0.649752i
106106 14105.3i 1.25536i
107107 1863.57 0.162771 0.0813857 0.996683i 0.474065π-0.474065\pi
0.0813857 + 0.996683i 0.474065π0.474065\pi
108108 6664.32i 0.571358i
109109 3764.74i 0.316870i −0.987369 0.158435i 0.949355π-0.949355\pi
0.987369 0.158435i 0.0506449π-0.0506449\pi
110110 −13289.5 6993.65i −1.09831 0.577988i
111111 3219.92i 0.261336i
112112 −15164.2 −1.20888
113113 −10.8182 −0.000847225 −0.000423612 1.00000i 0.500135π-0.500135\pi
−0.000423612 1.00000i 0.500135π0.500135\pi
114114 9883.14i 0.760475i
115115 −10387.0 + 8185.97i −0.785409 + 0.618977i
116116 6690.91 0.497244
117117 869.394i 0.0635104i
118118 20048.9i 1.43988i
119119 6208.12 0.438396
120120 −5633.86 2964.84i −0.391240 0.205892i
121121 −111.498 −0.00761547
122122 15758.3 1.05874
123123 12438.4i 0.822154i
124124 −3090.33 −0.200984
125125 1831.84 + 15517.2i 0.117238 + 0.993104i
126126 8061.18i 0.507759i
127127 16809.7i 1.04221i −0.853494 0.521103i 0.825521π-0.825521\pi
0.853494 0.521103i 0.174479π-0.174479\pi
128128 16973.7i 1.03599i
129129 48.6328i 0.00292247i
130130 −2767.39 1456.35i −0.163751 0.0861747i
131131 4191.57 0.244250 0.122125 0.992515i 0.461029π-0.461029\pi
0.122125 + 0.992515i 0.461029π0.461029\pi
132132 7015.89 0.402657
133133 13877.5i 0.784527i
134134 30139.1i 1.67850i
135135 9172.17 17429.2i 0.503274 0.956333i
136136 4882.48i 0.263975i
137137 8299.84 0.442210 0.221105 0.975250i 0.429034π-0.429034\pi
0.221105 + 0.975250i 0.429034π0.429034\pi
138138 7267.00 16319.8i 0.381590 0.856953i
139139 24420.7 1.26395 0.631973 0.774990i 0.282245π-0.282245\pi
0.631973 + 0.774990i 0.282245π0.282245\pi
140140 −8874.47 4670.22i −0.452779 0.238277i
141141 −17345.7 −0.872476
142142 21701.1i 1.07623i
143143 −3072.05 −0.150230
144144 −10992.2 −0.530103
145145 −17498.7 9208.77i −0.832282 0.437992i
146146 15910.7 0.746421
147147 1040.71i 0.0481611i
148148 3988.97 0.182112
149149 104.420i 0.00470339i −0.999997 0.00235170i 0.999251π-0.999251\pi
0.999997 0.00235170i 0.000748569π-0.000748569\pi
150150 −11951.4 17396.9i −0.531175 0.773197i
151151 −27841.9 −1.22108 −0.610542 0.791984i 0.709048π-0.709048\pi
−0.610542 + 0.791984i 0.709048π0.709048\pi
152152 10914.2 0.472393
153153 4500.15 0.192240
154154 −28484.6 −1.20107
155155 8082.13 + 4253.25i 0.336405 + 0.177034i
156156 1460.98 0.0600338
157157 −46395.1 −1.88223 −0.941116 0.338084i 0.890221π-0.890221\pi
−0.941116 + 0.338084i 0.890221π0.890221\pi
158158 43643.1 1.74824
159159 19475.0i 0.770340i
160160 −11466.3 + 21788.6i −0.447903 + 0.851117i
161161 −10204.0 + 22915.6i −0.393659 + 0.884056i
162162 12835.1i 0.489066i
163163 29520.0i 1.11107i 0.831493 + 0.555535i 0.187486π0.187486\pi
−0.831493 + 0.555535i 0.812514π0.812514\pi
164164 15409.2 0.572917
165165 −18348.7 9656.05i −0.673963 0.354676i
166166 63610.9i 2.30842i
167167 37239.6i 1.33528i −0.744485 0.667639i 0.767305π-0.767305\pi
0.744485 0.667639i 0.232695π-0.232695\pi
168168 −12075.6 −0.427847
169169 27921.3 0.977602
170170 7538.35 14324.6i 0.260843 0.495659i
171171 10059.5i 0.344022i
172172 60.2484 0.00203652
173173 19212.6i 0.641940i 0.947089 + 0.320970i 0.104009π0.104009\pi
−0.947089 + 0.320970i 0.895991π0.895991\pi
174174 26711.1 0.882252
175175 16781.7 + 24428.1i 0.547974 + 0.797651i
176176 38841.5i 1.25392i
177177 27681.3i 0.883569i
178178 27154.8 0.857051
179179 54673.8 1.70637 0.853185 0.521609i 0.174668π-0.174668\pi
0.853185 + 0.521609i 0.174668π0.174668\pi
180180 −6432.94 3385.36i −0.198547 0.104486i
181181 60172.9i 1.83672i 0.395742 + 0.918362i 0.370488π0.370488\pi
−0.395742 + 0.918362i 0.629512π0.629512\pi
182182 −5931.60 −0.179073
183183 21757.3 0.649685
184184 18022.3 + 8025.12i 0.532324 + 0.237037i
185185 −10432.3 5490.06i −0.304817 0.160411i
186186 −12337.0 −0.356603
187187 15901.5i 0.454731i
188188 21488.6i 0.607984i
189189 37357.5i 1.04581i
190190 32020.8 + 16851.1i 0.887002 + 0.466788i
191191 48565.0i 1.33124i 0.746290 + 0.665621i 0.231833π0.231833\pi
−0.746290 + 0.665621i 0.768167π0.768167\pi
192192 1678.87i 0.0455424i
193193 22851.5i 0.613479i 0.951793 + 0.306740i 0.0992381π0.0992381\pi
−0.951793 + 0.306740i 0.900762π0.900762\pi
194194 76759.3i 2.03952i
195195 −3820.91 2010.77i −0.100484 0.0528801i
196196 1289.28 0.0335610
197197 14182.4i 0.365441i −0.983165 0.182720i 0.941510π-0.941510\pi
0.983165 0.182720i 0.0584903π-0.0584903\pi
198198 −20647.9 −0.526679
199199 44550.7i 1.12499i 0.826801 + 0.562495i 0.190159π0.190159\pi
−0.826801 + 0.562495i 0.809841π0.809841\pi
200200 19211.8 13198.2i 0.480296 0.329956i
201201 41612.7i 1.02999i
202202 15068.1i 0.369279i
203203 −37506.6 −0.910155
204204 7562.33i 0.181717i
205205 −40299.6 21207.8i −0.958943 0.504647i
206206 30712.8i 0.723745i
207207 −7396.71 + 16611.1i −0.172623 + 0.387666i
208208 8088.33i 0.186953i
209209 35545.8 0.813760
210210 −35428.1 18644.2i −0.803359 0.422771i
211211 −41236.0 −0.926215 −0.463108 0.886302i 0.653266π-0.653266\pi
−0.463108 + 0.886302i 0.653266π0.653266\pi
212212 −24126.4 −0.536810
213213 29962.5i 0.660417i
214214 9216.53i 0.201252i
215215 −157.567 82.9205i −0.00340871 0.00179384i
216216 −29380.4 −0.629724
217217 17323.1 0.367881
218218 −18619.0 −0.391781
219219 21967.7 0.458033
220220 −11962.3 + 22731.1i −0.247155 + 0.469650i
221221 3311.31i 0.0677978i
222222 15924.5 0.323118
223223 15976.4i 0.321269i −0.987014 0.160634i 0.948646π-0.948646\pi
0.987014 0.160634i 0.0513540π-0.0513540\pi
224224 46701.4i 0.930752i
225225 12164.7 + 17707.4i 0.240291 + 0.349776i
226226 53.5029i 0.00104752i
227227 48395.1 0.939181 0.469590 0.882884i 0.344402π-0.344402\pi
0.469590 + 0.882884i 0.344402π0.344402\pi
228228 −16904.7 −0.325190
229229 69898.5i 1.33290i −0.745551 0.666449i 0.767814π-0.767814\pi
0.745551 0.666449i 0.232186π-0.232186\pi
230230 40484.8 + 51370.5i 0.765308 + 0.971086i
231231 −39328.3 −0.737023
232232 29497.6i 0.548039i
233233 64869.9i 1.19490i −0.801906 0.597450i 0.796181π-0.796181\pi
0.801906 0.597450i 0.203819π-0.203819\pi
234234 −4299.70 −0.0785248
235235 29575.0 56199.0i 0.535536 1.01764i
236236 34292.8 0.615714
237237 60257.5 1.07279
238238 30703.1i 0.542036i
239239 −29088.5 −0.509243 −0.254621 0.967041i 0.581951π-0.581951\pi
−0.254621 + 0.967041i 0.581951π0.581951\pi
240240 −25423.2 + 48309.8i −0.441375 + 0.838711i
241241 60948.1i 1.04936i −0.851298 0.524682i 0.824184π-0.824184\pi
0.851298 0.524682i 0.175816π-0.175816\pi
242242 551.429i 0.00941583i
243243 46091.6i 0.780565i
244244 26953.8i 0.452732i
245245 −3371.85 1774.45i −0.0561741 0.0295618i
246246 61515.6 1.01652
247247 7402.03 0.121327
248248 13624.1i 0.221515i
249249 87826.7i 1.41654i
250250 76742.6 9059.61i 1.22788 0.144954i
251251 21475.3i 0.340873i −0.985369 0.170437i 0.945482π-0.945482\pi
0.985369 0.170437i 0.0545178π-0.0545178\pi
252252 −13788.3 −0.217125
253253 58696.1 + 26136.6i 0.916997 + 0.408327i
254254 −83134.8 −1.28859
255255 10408.1 19777.7i 0.160063 0.304156i
256256 80012.0 1.22089
257257 89616.4i 1.35682i 0.734685 + 0.678409i 0.237330π0.237330\pi
−0.734685 + 0.678409i 0.762670π0.762670\pi
258258 240.520 0.00361336
259259 −22360.6 −0.333337
260260 −2491.02 + 4733.50i −0.0368494 + 0.0700222i
261261 −27187.8 −0.399111
262262 20730.0i 0.301992i
263263 21291.6 0.307820 0.153910 0.988085i 0.450813π-0.450813\pi
0.153910 + 0.988085i 0.450813π0.450813\pi
264264 30930.3i 0.443789i
265265 63097.7 + 33205.4i 0.898508 + 0.472843i
266266 68633.0 0.969995
267267 37492.3 0.525919
268268 −51551.5 −0.717748
269269 −117290. −1.62090 −0.810450 0.585808i 0.800777π-0.800777\pi
−0.810450 + 0.585808i 0.800777π0.800777\pi
270270 −86198.3 45362.2i −1.18242 0.622252i
271271 −110364. −1.50276 −0.751379 0.659871i 0.770611π-0.770611\pi
−0.751379 + 0.659871i 0.770611π0.770611\pi
272272 −41866.7 −0.565889
273273 −8189.69 −0.109886
274274 41048.0i 0.546752i
275275 62570.1 42984.7i 0.827373 0.568393i
276276 −27914.3 12429.9i −0.366445 0.163173i
277277 12201.9i 0.159025i −0.996834 0.0795127i 0.974664π-0.974664\pi
0.996834 0.0795127i 0.0253364π-0.0253364\pi
278278 120776.i 1.56275i
279279 12557.2 0.161319
280280 20589.2 39124.1i 0.262617 0.499032i
281281 138763.i 1.75737i −0.477405 0.878683i 0.658423π-0.658423\pi
0.477405 0.878683i 0.341577π-0.341577\pi
282282 85785.4i 1.07874i
283283 −2905.51 −0.0362785 −0.0181392 0.999835i 0.505774π-0.505774\pi
−0.0181392 + 0.999835i 0.505774π0.505774\pi
284284 −37118.8 −0.460211
285285 44210.7 + 23266.0i 0.544299 + 0.286439i
286286 15193.2i 0.185745i
287287 −86377.6 −1.04867
288288 33853.0i 0.408142i
289289 −66381.0 −0.794782
290290 −45543.2 + 86542.3i −0.541536 + 1.02904i
291291 105981.i 1.25153i
292292 27214.5i 0.319180i
293293 −96882.7 −1.12852 −0.564262 0.825596i 0.690839π-0.690839\pi
−0.564262 + 0.825596i 0.690839π0.690839\pi
294294 5146.99 0.0595468
295295 −89685.9 47197.5i −1.03058 0.542345i
296296 17585.8i 0.200715i
297297 −95687.5 −1.08478
298298 −516.423 −0.00581531
299299 12222.8 + 5442.66i 0.136719 + 0.0608792i
300300 −29756.6 + 20442.4i −0.330629 + 0.227137i
301301 −337.728 −0.00372764
302302 137696.i 1.50976i
303303 20804.3i 0.226604i
304304 93587.9i 1.01268i
305305 −37096.9 + 70492.3i −0.398784 + 0.757778i
306306 22256.1i 0.237687i
307307 80278.2i 0.851767i −0.904778 0.425884i 0.859963π-0.859963\pi
0.904778 0.425884i 0.140037π-0.140037\pi
308308 48721.5i 0.513594i
309309 42404.8i 0.444118i
310310 21035.0 39971.2i 0.218887 0.415934i
311311 99847.5 1.03233 0.516163 0.856491i 0.327360π-0.327360\pi
0.516163 + 0.856491i 0.327360π0.327360\pi
312312 6440.91i 0.0661664i
313313 9836.64 0.100406 0.0502028 0.998739i 0.484013π-0.484013\pi
0.0502028 + 0.998739i 0.484013π0.484013\pi
314314 229453.i 2.32721i
315315 36060.5 + 18977.0i 0.363421 + 0.191252i
316316 74649.5i 0.747572i
317317 104833.i 1.04323i −0.853182 0.521613i 0.825330π-0.825330\pi
0.853182 0.521613i 0.174670π-0.174670\pi
318318 −96316.0 −0.952454
319319 96069.4i 0.944069i
320320 −5439.45 2862.53i −0.0531196 0.0279544i
321321 12725.1i 0.123496i
322322 113332. + 50465.4i 1.09305 + 0.486723i
323323 38314.3i 0.367245i
324324 21953.8 0.209131
325325 13029.5 8951.09i 0.123356 0.0847441i
326326 145995. 1.37374
327327 −25707.0 −0.240412
328328 67933.0i 0.631442i
329329 120456.i 1.11285i
330330 −47755.3 + 90745.7i −0.438524 + 0.833294i
331331 −38032.9 −0.347139 −0.173569 0.984822i 0.555530π-0.555530\pi
−0.173569 + 0.984822i 0.555530π0.555530\pi
332332 −108803. −0.987112
333333 −16208.8 −0.146171
334334 −184173. −1.65095
335335 134823. + 70950.9i 1.20136 + 0.632220i
336336 103547.i 0.917186i
337337 −33040.3 −0.290927 −0.145463 0.989364i 0.546467π-0.546467\pi
−0.145463 + 0.989364i 0.546467π0.546467\pi
338338 138088.i 1.20871i
339339 73.8708i 0.000642796i
340340 −24501.5 12894.0i −0.211951 0.111540i
341341 44371.5i 0.381589i
342342 49750.8 0.425351
343343 −121081. −1.02917
344344 265.612i 0.00224456i
345345 55896.8 + 70926.6i 0.469622 + 0.595896i
346346 95018.6 0.793700
347347 43160.3i 0.358448i −0.983808 0.179224i 0.942641π-0.942641\pi
0.983808 0.179224i 0.0573586π-0.0573586\pi
348348 45688.0i 0.377263i
349349 −91666.3 −0.752591 −0.376296 0.926500i 0.622802π-0.622802\pi
−0.376296 + 0.926500i 0.622802π0.622802\pi
350350 120812. 82996.1i 0.986222 0.677519i
351351 −19925.9 −0.161735
352352 119621. 0.965433
353353 51870.6i 0.416267i 0.978100 + 0.208133i 0.0667388π0.0667388\pi
−0.978100 + 0.208133i 0.933261π0.933261\pi
354354 136902. 1.09245
355355 97076.6 + 51086.9i 0.770296 + 0.405371i
356356 46447.0i 0.366486i
357357 42391.4i 0.332614i
358358 270396.i 2.10977i
359359 240529.i 1.86629i 0.359499 + 0.933146i 0.382948π0.382948\pi
−0.359499 + 0.933146i 0.617052π0.617052\pi
360360 14924.7 28360.3i 0.115160 0.218830i
361361 44674.0 0.342800
362362 297593. 2.27094
363363 761.351i 0.00577792i
364364 10145.7i 0.0765738i
365365 −37455.7 + 71174.1i −0.281146 + 0.534240i
366366 107604.i 0.803275i
367367 −188291. −1.39797 −0.698983 0.715139i 0.746363π-0.746363\pi
−0.698983 + 0.715139i 0.746363π0.746363\pi
368368 68814.5 154540.i 0.508141 1.14115i
369369 −62613.5 −0.459849
370370 −27151.8 + 51594.5i −0.198333 + 0.376878i
371371 135243. 0.982577
372372 21101.9i 0.152488i
373373 156696. 1.12626 0.563132 0.826367i 0.309596π-0.309596\pi
0.563132 + 0.826367i 0.309596π0.309596\pi
374374 −78643.0 −0.562233
375375 105958. 12508.5i 0.753476 0.0889493i
376376 −94734.8 −0.670091
377377 20005.4i 0.140755i
378378 −184756. −1.29305
379379 148654.i 1.03490i 0.855714 + 0.517449i 0.173118π0.173118\pi
−0.855714 + 0.517449i 0.826882π0.826882\pi
380380 28823.0 54770.1i 0.199605 0.379294i
381381 −114783. −0.790730
382382 240185. 1.64596
383383 122905. 0.837859 0.418929 0.908019i 0.362405π-0.362405\pi
0.418929 + 0.908019i 0.362405π0.362405\pi
384384 115903. 0.786017
385385 67056.0 127421.i 0.452393 0.859648i
386386 113015. 0.758511
387387 −244.813 −0.00163460
388388 131293. 0.872125
389389 122657.i 0.810578i −0.914189 0.405289i 0.867171π-0.867171\pi
0.914189 0.405289i 0.132829π-0.132829\pi
390390 −9944.51 + 18896.8i −0.0653814 + 0.124239i
391391 −28172.3 + 63267.6i −0.184276 + 0.413836i
392392 5683.94i 0.0369894i
393393 28621.6i 0.185314i
394394 −70140.9 −0.451834
395395 −102741. + 195231.i −0.658490 + 1.25128i
396396 35317.3i 0.225215i
397397 99402.5i 0.630691i −0.948977 0.315345i 0.897880π-0.897880\pi
0.948977 0.315345i 0.102120π-0.102120\pi
398398 220332. 1.39095
399399 94760.7 0.595227
400400 −113173. 164739.i −0.707334 1.02962i
401401 233065.i 1.44940i 0.689065 + 0.724700i 0.258021π0.258021\pi
−0.689065 + 0.724700i 0.741979π0.741979\pi
402402 −205801. −1.27349
403403 9239.88i 0.0568927i
404404 −25773.2 −0.157909
405405 −57415.7 30215.2i −0.350042 0.184211i
406406 185494.i 1.12532i
407407 57274.4i 0.345758i
408408 −33339.4 −0.200280
409409 92352.0 0.552077 0.276038 0.961147i 0.410978π-0.410978\pi
0.276038 + 0.961147i 0.410978π0.410978\pi
410410 −104886. + 199307.i −0.623950 + 1.18564i
411411 56674.4i 0.335508i
412412 52532.9 0.309483
413413 −192232. −1.12700
414414 82152.4 + 36581.4i 0.479313 + 0.213432i
415415 284553. + 149747.i 1.65222 + 0.869486i
416416 24909.8 0.143941
417417 166754.i 0.958966i
418418 175797.i 1.00614i
419419 64162.6i 0.365472i −0.983162 0.182736i 0.941505π-0.941505\pi
0.983162 0.182736i 0.0584953π-0.0584953\pi
420420 −31890.0 + 60598.2i −0.180782 + 0.343527i
421421 125400.i 0.707514i −0.935337 0.353757i 0.884904π-0.884904\pi
0.935337 0.353757i 0.115096π-0.115096\pi
422422 203938.i 1.14518i
423423 87316.5i 0.487995i
424424 106364.i 0.591647i
425425 46332.5 + 67443.3i 0.256512 + 0.373389i
426426 −148183. −0.816545
427427 151092.i 0.828680i
428428 −15764.4 −0.0860580
429429 20977.1i 0.113980i
430430 −410.094 + 779.271i −0.00221792 + 0.00421455i
431431 202228.i 1.08864i −0.838876 0.544322i 0.816787π-0.816787\pi
0.838876 0.544322i 0.183213π-0.183213\pi
432432 251934.i 1.34995i
433433 −231136. −1.23280 −0.616400 0.787433i 0.711409π-0.711409\pi
−0.616400 + 0.787433i 0.711409π0.711409\pi
434434 85673.9i 0.454851i
435435 −62880.9 + 119488.i −0.332308 + 0.631459i
436436 31847.0i 0.167531i
437437 −141427. 62975.6i −0.740576 0.329769i
438438 108644.i 0.566316i
439439 282237. 1.46449 0.732243 0.681044i 0.238474π-0.238474\pi
0.732243 + 0.681044i 0.238474π0.238474\pi
440440 −100213. 52737.2i −0.517627 0.272403i
441441 −5238.85 −0.0269376
442442 −16376.5 −0.0838258
443443 123666.i 0.630147i −0.949067 0.315073i 0.897971π-0.897971\pi
0.949067 0.315073i 0.102029π-0.102029\pi
444444 27238.2i 0.138169i
445445 −63925.5 + 121473.i −0.322815 + 0.613421i
446446 −79013.2 −0.397219
447447 −713.019 −0.00356850
448448 −11658.9 −0.0580898
449449 −258480. −1.28214 −0.641069 0.767483i 0.721509π-0.721509\pi
−0.641069 + 0.767483i 0.721509π0.721509\pi
450450 87574.4 60162.3i 0.432466 0.297098i
451451 221248.i 1.08774i
452452 91.5142 0.000447932
453453 190115.i 0.926445i
454454 239344.i 1.16121i
455455 13963.7 26534.1i 0.0674492 0.128169i
456456 74526.0i 0.358409i
457457 −184081. −0.881407 −0.440703 0.897653i 0.645271π-0.645271\pi
−0.440703 + 0.897653i 0.645271π0.645271\pi
458458 −345692. −1.64800
459459 103140.i 0.489556i
460460 87866.8 69247.3i 0.415250 0.327256i
461461 −39428.1 −0.185526 −0.0927628 0.995688i 0.529570π-0.529570\pi
−0.0927628 + 0.995688i 0.529570π0.529570\pi
462462 194503.i 0.911261i
463463 157266.i 0.733623i 0.930295 + 0.366812i 0.119551π0.119551\pi
−0.930295 + 0.366812i 0.880449π0.880449\pi
464464 252939. 1.17484
465465 29042.8 55187.8i 0.134317 0.255233i
466466 −320823. −1.47738
467467 397293. 1.82170 0.910851 0.412735i 0.135426π-0.135426\pi
0.910851 + 0.412735i 0.135426π0.135426\pi
468468 7354.45i 0.0335783i
469469 288977. 1.31377
470470 −277940. 146267.i −1.25822 0.662140i
471471 316803.i 1.42806i
472472 151184.i 0.678611i
473473 865.058i 0.00386654i
474474 298011.i 1.32640i
475475 −150761. + 103571.i −0.668194 + 0.459039i
476476 −52516.2 −0.231782
477477 98035.0 0.430868
478478 143861.i 0.629632i
479479 137130.i 0.597668i 0.954305 + 0.298834i 0.0965977π0.0965977\pi
−0.954305 + 0.298834i 0.903402π0.903402\pi
480480 148781. + 78296.3i 0.645749 + 0.339828i
481481 11926.8i 0.0515505i
482482 −301427. −1.29744
483483 156476. + 69676.9i 0.670740 + 0.298672i
484484 943.193 0.00402634
485485 −343371. 180700.i −1.45975 0.768201i
486486 −227952. −0.965097
487487 333190.i 1.40487i −0.711750 0.702433i 0.752097π-0.752097\pi
0.711750 0.702433i 0.247903π-0.247903\pi
488488 118829. 0.498980
489489 201574. 0.842977
490490 −8775.78 + 16675.9i −0.0365505 + 0.0694541i
491491 72764.5 0.301826 0.150913 0.988547i 0.451779π-0.451779\pi
0.150913 + 0.988547i 0.451779π0.451779\pi
492492 105220.i 0.434676i
493493 −103552. −0.426053
494494 36607.7i 0.150010i
495495 48607.6 92365.3i 0.198378 0.376963i
496496 −116825. −0.474867
497497 208073. 0.842369
498498 −434359. −1.75142
499499 −64799.7 −0.260239 −0.130119 0.991498i 0.541536π-0.541536\pi
−0.130119 + 0.991498i 0.541536π0.541536\pi
500500 −15496.0 131265.i −0.0619842 0.525059i
501501 −254286. −1.01309
502502 −106209. −0.421458
503503 66136.2 0.261399 0.130699 0.991422i 0.458278π-0.458278\pi
0.130699 + 0.991422i 0.458278π0.458278\pi
504504 60787.2i 0.239304i
505505 67404.7 + 35472.0i 0.264306 + 0.139092i
506506 129262. 290289.i 0.504859 1.13378i
507507 190657.i 0.741714i
508508 142198.i 0.551019i
509509 −278831. −1.07623 −0.538115 0.842871i 0.680864π-0.680864\pi
−0.538115 + 0.842871i 0.680864π0.680864\pi
510510 −97813.4 51474.7i −0.376061 0.197903i
511511 152554.i 0.584227i
512512 124130.i 0.473518i
513513 230557. 0.876080
514514 443210. 1.67758
515515 −137389. 72301.6i −0.518010 0.272605i
516516 411.398i 0.00154512i
517517 −308537. −1.15432
518518 110587.i 0.412140i
519519 131191. 0.487045
520520 −20868.2 10981.9i −0.0771751 0.0406137i
521521 335750.i 1.23692i −0.785817 0.618459i 0.787757π-0.787757\pi
0.785817 0.618459i 0.212243π-0.212243\pi
522522 134461.i 0.493463i
523523 343493. 1.25578 0.627891 0.778301i 0.283918π-0.283918\pi
0.627891 + 0.778301i 0.283918π0.283918\pi
524524 −35457.6 −0.129136
525525 166804. 114592.i 0.605184 0.415752i
526526 105301.i 0.380592i
527527 47827.4 0.172209
528528 265224. 0.951362
529529 −187230. 207981.i −0.669058 0.743210i
530530 164222. 312058.i 0.584627 1.11092i
531531 −139345. −0.494200
532532 117394.i 0.414783i
533533 46072.4i 0.162176i
534534 185423.i 0.650251i
535535 41228.7 + 21696.8i 0.144043 + 0.0758032i
536536 227271.i 0.791068i
537537 373333.i 1.29464i
538538 580073.i 2.00409i
539539 18511.7i 0.0637191i
540540 −77589.9 + 147438.i −0.266083 + 0.505618i
541541 317068. 1.08332 0.541661 0.840597i 0.317796π-0.317796\pi
0.541661 + 0.840597i 0.317796π0.317796\pi
542542 545820.i 1.85802i
543543 410883. 1.39354
544544 128938.i 0.435695i
545545 43831.3 83289.3i 0.147568 0.280412i
546546 40503.2i 0.135864i
547547 5272.95i 0.0176230i 0.999961 + 0.00881148i 0.00280482π0.00280482\pi
−0.999961 + 0.00881148i 0.997195π0.997195\pi
548548 −70210.6 −0.233799
549549 109524.i 0.363383i
550550 −212586. 309448.i −0.702765 1.02297i
551551 231477.i 0.762439i
552552 54798.5 123063.i 0.179842 0.403878i
553553 418455.i 1.36835i
554554 −60345.9 −0.196620
555555 −37488.2 + 71235.9i −0.121705 + 0.231267i
556556 −206582. −0.668254
557557 279435. 0.900679 0.450340 0.892857i 0.351303π-0.351303\pi
0.450340 + 0.892857i 0.351303π0.351303\pi
558558 62103.4i 0.199456i
559559 180.139i 0.000576479i
560560 −335485. 176550.i −1.06979 0.562979i
561561 −108581. −0.345008
562562 −686273. −2.17282
563563 −259566. −0.818902 −0.409451 0.912332i 0.634280π-0.634280\pi
−0.409451 + 0.912332i 0.634280π0.634280\pi
564564 146732. 0.461282
565565 −239.337 125.952i −0.000749744 0.000394556i
566566 14369.6i 0.0448550i
567567 −123064. −0.382794
568568 163642.i 0.507223i
569569 298678.i 0.922527i −0.887263 0.461264i 0.847396π-0.847396\pi
0.887263 0.461264i 0.152604π-0.152604\pi
570570 115065. 218650.i 0.354156 0.672976i
571571 335764.i 1.02982i 0.857244 + 0.514910i 0.172175π0.172175\pi
−0.857244 + 0.514910i 0.827825π0.827825\pi
572572 25987.3 0.0794271
573573 331620. 1.01002
574574 427192.i 1.29658i
575575 −325104. + 60170.4i −0.983300 + 0.181990i
576576 −8451.29 −0.0254729
577577 159733.i 0.479782i 0.970800 + 0.239891i 0.0771117π0.0771117\pi
−0.970800 + 0.239891i 0.922888π0.922888\pi
578578 328296.i 0.982675i
579579 156038. 0.465452
580580 148027. + 77899.5i 0.440031 + 0.231568i
581581 609908. 1.80681
582582 524141. 1.54740
583583 346411.i 1.01919i
584584 119978. 0.351785
585585 10122.0 19234.1i 0.0295770 0.0562030i
586586 479146.i 1.39532i
587587 524199.i 1.52132i −0.649151 0.760659i 0.724876π-0.724876\pi
0.649151 0.760659i 0.275124π-0.275124\pi
588588 8803.68i 0.0254630i
589589 106912.i 0.308175i
590590 −233422. + 443553.i −0.670559 + 1.27421i
591591 −96842.7 −0.277263
592592 150797. 0.430277
593593 272680.i 0.775432i −0.921779 0.387716i 0.873264π-0.873264\pi
0.921779 0.387716i 0.126736π-0.126736\pi
594594 473235.i 1.34123i
595595 137346. + 72278.6i 0.387954 + 0.204162i
596596 883.318i 0.00248671i
597597 304209. 0.853539
598598 26917.4 60449.6i 0.0752716 0.169041i
599599 −372361. −1.03779 −0.518896 0.854838i 0.673657π-0.673657\pi
−0.518896 + 0.854838i 0.673657π0.673657\pi
600600 −90122.4 131185.i −0.250340 0.364404i
601601 686200. 1.89977 0.949887 0.312594i 0.101198π-0.101198\pi
0.949887 + 0.312594i 0.101198π0.101198\pi
602602 1670.28i 0.00460889i
603603 209474. 0.576097
604604 235522. 0.645592
605605 −2466.73 1298.13i −0.00673924 0.00354655i
606606 −102890. −0.280175
607607 39449.6i 0.107069i −0.998566 0.0535346i 0.982951π-0.982951\pi
0.998566 0.0535346i 0.0170488π-0.0170488\pi
608608 −288224. −0.779693
609609 256109.i 0.690542i
610610 348629. + 183467.i 0.936923 + 0.493059i
611611 −64249.5 −0.172102
612612 −38068.0 −0.101638
613613 −328168. −0.873323 −0.436662 0.899626i 0.643839π-0.643839\pi
−0.436662 + 0.899626i 0.643839π0.643839\pi
614614 −397026. −1.05313
615615 −144815. + 275180.i −0.382880 + 0.727557i
616616 −214794. −0.566059
617617 −676661. −1.77746 −0.888731 0.458428i 0.848413π-0.848413\pi
−0.888731 + 0.458428i 0.848413π0.848413\pi
618618 209719. 0.549111
619619 334267.i 0.872394i 0.899851 + 0.436197i 0.143675π0.143675\pi
−0.899851 + 0.436197i 0.856325π0.856325\pi
620620 −68368.9 35979.4i −0.177859 0.0935989i
621621 380714. + 169527.i 0.987224 + 0.439598i
622622 493809.i 1.27638i
623623 260363.i 0.670816i
624624 55230.1 0.141843
625625 −140134. + 364624.i −0.358743 + 0.933436i
626626 48648.4i 0.124142i
627627 242720.i 0.617406i
628628 392469. 0.995145
629629 −61735.2 −0.156038
630630 93853.0 178342.i 0.236465 0.449337i
631631 331904.i 0.833592i 0.909000 + 0.416796i 0.136847π0.136847\pi
−0.909000 + 0.416796i 0.863153π0.863153\pi
632632 329101. 0.823938
633633 281575.i 0.702727i
634634 −518464. −1.28985
635635 195709. 371891.i 0.485359 0.922290i
636636 164744.i 0.407282i
637637 3854.87i 0.00950015i
638638 475124. 1.16725
639639 150828. 0.369386
640640 −197618. + 375519.i −0.482466 + 0.916794i
641641 802258.i 1.95253i −0.216579 0.976265i 0.569490π-0.569490\pi
0.216579 0.976265i 0.430510π-0.430510\pi
642642 −62933.9 −0.152691
643643 −143766. −0.347724 −0.173862 0.984770i 0.555625π-0.555625\pi
−0.173862 + 0.984770i 0.555625π0.555625\pi
644644 86318.7 193849.i 0.208129 0.467405i
645645 −566.212 + 1075.93i −0.00136100 + 0.00258621i
646646 189489. 0.454065
647647 756331.i 1.80677i 0.428828 + 0.903386i 0.358927π0.358927\pi
−0.428828 + 0.903386i 0.641073π0.641073\pi
648648 96785.7i 0.230495i
649649 492382.i 1.16900i
650650 −44268.8 64439.3i −0.104778 0.152519i
651651 118289.i 0.279114i
652652 249718.i 0.587428i
653653 442957.i 1.03881i 0.854529 + 0.519404i 0.173846π0.173846\pi
−0.854529 + 0.519404i 0.826154π0.826154\pi
654654 127137.i 0.297247i
655655 92732.3 + 48800.7i 0.216147 + 0.113748i
656656 582518. 1.35364
657657 110583.i 0.256188i
658658 −595733. −1.37594
659659 654514.i 1.50712i −0.657379 0.753560i 0.728335π-0.728335\pi
0.657379 0.753560i 0.271665π-0.271665\pi
660660 155216. + 81683.2i 0.356328 + 0.187519i
661661 175823.i 0.402413i 0.979549 + 0.201206i 0.0644862π0.0644862\pi
−0.979549 + 0.201206i 0.935514π0.935514\pi
662662 188097.i 0.429205i
663663 −22610.9 −0.0514387
664664 479672.i 1.08795i
665665 −161570. + 307019.i −0.365357 + 0.694260i
666666 80162.5i 0.180727i
667667 170204. 382233.i 0.382576 0.859166i
668668 315020.i 0.705968i
669669 −109093. −0.243749
670670 350897. 666783.i 0.781682 1.48537i
671671 387008. 0.859558
672672 318894. 0.706169
673673 487162.i 1.07558i 0.843078 + 0.537791i 0.180741π0.180741\pi
−0.843078 + 0.537791i 0.819259π0.819259\pi
674674 163405.i 0.359704i
675675 405841. 278807.i 0.890736 0.611922i
676676 −236194. −0.516863
677677 −372718. −0.813210 −0.406605 0.913604i 0.633288π-0.633288\pi
−0.406605 + 0.913604i 0.633288π0.633288\pi
678678 365.338 0.000794758
679679 −735977. −1.59634
680680 56844.7 108018.i 0.122934 0.233602i
681681 330459.i 0.712564i
682682 −219445. −0.471799
683683 278036.i 0.596019i −0.954563 0.298009i 0.903677π-0.903677\pi
0.954563 0.298009i 0.0963227π-0.0963227\pi
684684 85096.4i 0.181886i
685685 183622. + 96631.6i 0.391330 + 0.205939i
686686 598822.i 1.27248i
687687 −477293. −1.01128
688688 2277.59 0.00481171
689689 72136.4i 0.151955i
690690 350777. 276445.i 0.736771 0.580645i
691691 −335596. −0.702848 −0.351424 0.936216i 0.614302π-0.614302\pi
−0.351424 + 0.936216i 0.614302π0.614302\pi
692692 162525.i 0.339397i
693693 197975.i 0.412234i
694694 −213455. −0.443188
695695 540272. + 284320.i 1.11852 + 0.588624i
696696 201421. 0.415801
697697 −238480. −0.490892
698698 453348.i 0.930510i
699699 −442956. −0.906580
700700 −141961. 206644.i −0.289716 0.421722i
701701 220951.i 0.449635i −0.974401 0.224817i 0.927821π-0.927821\pi
0.974401 0.224817i 0.0721785π-0.0721785\pi
702702 98546.1i 0.199970i
703703 138001.i 0.279237i
704704 29863.0i 0.0602543i
705705 −383748. 201949.i −0.772090 0.406315i
706706 256533. 0.514675
707707 144474. 0.289036
708708 234164.i 0.467147i
709709 755284.i 1.50251i 0.660012 + 0.751255i 0.270551π0.270551\pi
−0.660012 + 0.751255i 0.729449π0.729449\pi
710710 252657. 480105.i 0.501204 0.952400i
711711 303330.i 0.600035i
712712 204767. 0.403924
713713 −78611.9 + 176542.i −0.154635 + 0.347271i
714714 −209652. −0.411247
715715 −67964.5 35766.6i −0.132944 0.0699625i
716716 −462501. −0.902165
717717 198627.i 0.386367i
718718 1.18957e6 2.30750
719719 −68333.1 −0.132182 −0.0660912 0.997814i 0.521053π-0.521053\pi
−0.0660912 + 0.997814i 0.521053π0.521053\pi
720720 −243187. 127978.i −0.469110 0.246871i
721721 −294478. −0.566478
722722 220941.i 0.423841i
723723 −416176. −0.796160
724724 509019.i 0.971084i
725725 −279920. 407461.i −0.532546 0.775193i
726726 3765.36 0.00714387
727727 −343164. −0.649281 −0.324640 0.945838i 0.605243π-0.605243\pi
−0.324640 + 0.945838i 0.605243π0.605243\pi
728728 −44728.6 −0.0843961
729729 −524944. −0.987775
730730 352001. + 185242.i 0.660539 + 0.347611i
731731 −932.433 −0.00174495
732732 −184051. −0.343491
733733 449850. 0.837259 0.418630 0.908157i 0.362511π-0.362511\pi
0.418630 + 0.908157i 0.362511π0.362511\pi
734734 931216.i 1.72846i
735735 −12116.6 + 23024.3i −0.0224288 + 0.0426197i
736736 −475939. 211929.i −0.878609 0.391233i
737737 740186.i 1.36272i
738738 309663.i 0.568561i
739739 321903. 0.589435 0.294717 0.955584i 0.404774π-0.404774\pi
0.294717 + 0.955584i 0.404774π0.404774\pi
740740 88250.1 + 46441.9i 0.161158 + 0.0848100i
741741 50543.8i 0.0920517i
742742 668862.i 1.21487i
743743 −725883. −1.31489 −0.657445 0.753503i 0.728363π-0.728363\pi
−0.657445 + 0.753503i 0.728363π0.728363\pi
744744 −93030.1 −0.168065
745745 1215.72 2310.14i 0.00219039 0.00416222i
746746 774960.i 1.39252i
747747 442111. 0.792301
748748 134515.i 0.240419i
749749 88369.1 0.157520
750750 −61862.4 524027.i −0.109978 0.931603i
751751 943161.i 1.67227i −0.548525 0.836134i 0.684810π-0.684810\pi
0.548525 0.836134i 0.315190π-0.315190\pi
752752 812341.i 1.43649i
753753 −146642. −0.258623
754754 98939.4 0.174031
755755 −615961. 324152.i −1.08059 0.568663i
756756 316017.i 0.552926i
757757 534635. 0.932965 0.466483 0.884530i 0.345521π-0.345521\pi
0.466483 + 0.884530i 0.345521π0.345521\pi
758758 735187. 1.27956
759759 178470. 400799.i 0.309801 0.695733i
760760 241460. + 127069.i 0.418040 + 0.219995i
761761 150078. 0.259148 0.129574 0.991570i 0.458639π-0.458639\pi
0.129574 + 0.991570i 0.458639π0.458639\pi
762762 567675.i 0.977665i
763763 178521.i 0.306648i
764764 410825.i 0.703834i
765765 99559.2 + 52393.4i 0.170121 + 0.0895269i
766766 607841.i 1.03594i
767767 102533.i 0.174291i
768768 546351.i 0.926296i
769769 238646.i 0.403553i 0.979432 + 0.201777i 0.0646715π0.0646715\pi
−0.979432 + 0.201777i 0.935329π0.935329\pi
770770 −630179. 331634.i −1.06288 0.559342i
771771 611934. 1.02943
772772 193307.i 0.324349i
773773 197164. 0.329965 0.164983 0.986296i 0.447243π-0.447243\pi
0.164983 + 0.986296i 0.447243π0.447243\pi
774774 1210.75i 0.00202104i
775775 129286. + 188194.i 0.215253 + 0.313330i
776776 578821.i 0.961215i
777777 152686.i 0.252905i
778778 −606619. −1.00220
779779 533092.i 0.878471i
780780 32322.1 + 17009.6i 0.0531264 + 0.0279580i
781781 532958.i 0.873758i
782782 312898. + 139330.i 0.511670 + 0.227840i
783783 623125.i 1.01637i
784784 48739.1 0.0792950
785785 −1.02642e6 540159.i −1.66566 0.876562i
786786 −141552. −0.229124
787787 103089. 0.166441 0.0832206 0.996531i 0.473479π-0.473479\pi
0.0832206 + 0.996531i 0.473479π0.473479\pi
788788 119973.i 0.193210i
789789 145387.i 0.233546i
790790 965539. + 508119.i 1.54709 + 0.814162i
791791 −512.992 −0.000819894
792792 −155700. −0.248221
793793 80590.3 0.128155
794794 −491608. −0.779791
795795 226739. 430855.i 0.358750 0.681705i
796796 376867.i 0.594788i
797797 −1.05340e6 −1.65835 −0.829174 0.558990i 0.811189π-0.811189\pi
−0.829174 + 0.558990i 0.811189π0.811189\pi
798798 468651.i 0.735943i
799799 332568.i 0.520938i
800800 −507351. + 348542.i −0.792736 + 0.544598i
801801 188732.i 0.294159i
802802 1.15265e6 1.79205
803803 390752. 0.605996
804804 352013.i 0.544561i
805805 −492546. + 388173.i −0.760073 + 0.599009i
806806 −45697.1 −0.0703426
807807 800899.i 1.22979i
808808 113624.i 0.174040i
809809 −280562. −0.428679 −0.214340 0.976759i 0.568760π-0.568760\pi
−0.214340 + 0.976759i 0.568760π0.568760\pi
810810 −149433. + 283957.i −0.227760 + 0.432795i
811811 −977553. −1.48627 −0.743137 0.669140i 0.766663π-0.766663\pi
−0.743137 + 0.669140i 0.766663π0.766663\pi
812812 317278. 0.481203
813813 753607.i 1.14015i
814814 283258. 0.427497
815815 −343689. + 653086.i −0.517429 + 0.983231i
816816 285881.i 0.429344i
817817 2084.34i 0.00312266i
818818 456739.i 0.682592i
819819 41226.1i 0.0614616i
820820 340905. + 179403.i 0.506997 + 0.266809i
821821 240514. 0.356824 0.178412 0.983956i 0.442904π-0.442904\pi
0.178412 + 0.983956i 0.442904π0.442904\pi
822822 −280291. −0.414825
823823 148476.i 0.219208i −0.993975 0.109604i 0.965042π-0.965042\pi
0.993975 0.109604i 0.0349583π-0.0349583\pi
824824 231597.i 0.341098i
825825 −293515. 427252.i −0.431244 0.627734i
826826 950707.i 1.39343i
827827 −186234. −0.272301 −0.136150 0.990688i 0.543473π-0.543473\pi
−0.136150 + 0.990688i 0.543473π0.543473\pi
828828 62570.8 140518.i 0.0912664 0.204961i
829829 601485. 0.875217 0.437608 0.899166i 0.355826π-0.355826\pi
0.437608 + 0.899166i 0.355826π0.355826\pi
830830 740595. 1.40730e6i 1.07504 2.04282i
831831 −83318.9 −0.120654
832832 6218.65i 0.00898358i
833833 −19953.5 −0.0287561
834834 −824702. −1.18567
835835 433565. 823871.i 0.621844 1.18164i
836836 −300692. −0.430239
837837 287802.i 0.410812i
838838 −317325. −0.451872
839839 335094.i 0.476039i −0.971260 0.238020i 0.923502π-0.923502\pi
0.971260 0.238020i 0.0764982π-0.0764982\pi
840840 −267154. 140591.i −0.378619 0.199250i
841841 −81669.6 −0.115470
842842 −620184. −0.874775
843843 −947528. −1.33333
844844 348827. 0.489694
845845 617717. + 325076.i 0.865120 + 0.455272i
846846 −431835. −0.603361
847847 −5287.16 −0.00736980
848848 −912059. −1.26833
849849 19839.9i 0.0275248i
850850 333550. 229144.i 0.461661 0.317154i
851851 101472. 227879.i 0.140115 0.314663i
852852 253461.i 0.349165i
853853 374862.i 0.515197i −0.966252 0.257598i 0.917069π-0.917069\pi
0.966252 0.257598i 0.0829311π-0.0829311\pi
854854 747248. 1.02459
855855 −117119. + 222552.i −0.160212 + 0.304439i
856856 69499.3i 0.0948490i
857857 141026.i 0.192017i −0.995381 0.0960083i 0.969392π-0.969392\pi
0.995381 0.0960083i 0.0306075π-0.0306075\pi
858858 103745. 0.140926
859859 413213. 0.560000 0.280000 0.960000i 0.409666π-0.409666\pi
0.280000 + 0.960000i 0.409666π0.409666\pi
860860 1332.91 + 701.447i 0.00180220 + 0.000948414i
861861 589818.i 0.795632i
862862 −1.00014e6 −1.34601
863863 166710.i 0.223841i −0.993717 0.111920i 0.964300π-0.964300\pi
0.993717 0.111920i 0.0357002π-0.0357002\pi
864864 775885. 1.03937
865865 −223685. + 425051.i −0.298954 + 0.568079i
866866 1.14312e6i 1.52424i
867867 453274.i 0.603008i
868868 −146541. −0.194500
869869 1.07183e6 1.41934
870870 590943. + 310986.i 0.780741 + 0.410868i
871871 154136.i 0.203173i
872872 −140401. −0.184645
873873 −533496. −0.700007
874874 −311454. + 699446.i −0.407729 + 0.915654i
875875 86864.6 + 735817.i 0.113456 + 0.961067i
876876 −185831. −0.242164
877877 543966.i 0.707249i −0.935387 0.353625i 0.884949π-0.884949\pi
0.935387 0.353625i 0.115051π-0.115051\pi
878878 1.39584e6i 1.81070i
879879 661551.i 0.856220i
880880 −452216. + 859312.i −0.583957 + 1.10965i
881881 173103.i 0.223025i 0.993763 + 0.111512i 0.0355695π0.0355695\pi
−0.993763 + 0.111512i 0.964431π0.964431\pi
882882 25909.4i 0.0333059i
883883 350410.i 0.449423i 0.974425 + 0.224712i 0.0721440π0.0721440\pi
−0.974425 + 0.224712i 0.927856π0.927856\pi
884884 28011.3i 0.0358450i
885885 −322282. + 612408.i −0.411481 + 0.781906i
886886 −611605. −0.779118
887887 735051.i 0.934266i 0.884187 + 0.467133i 0.154713π0.154713\pi
−0.884187 + 0.467133i 0.845287π0.845287\pi
888888 120083. 0.152284
889889 797106.i 1.00859i
890890 600760. + 316152.i 0.758439 + 0.399131i
891891 315216.i 0.397058i
892892 135148.i 0.169856i
893893 743414. 0.932240
894894 3526.33i 0.00441212i
895895 1.20958e6 + 636544.i 1.51004 + 0.794662i
896896 804883.i 1.00257i
897897 37164.5 83461.9i 0.0461896 0.103730i
898898 1.27835e6i 1.58525i
899899 −288951. −0.357523
900900 −102905. 149792.i −0.127043 0.184928i
901901 373392. 0.459955
902902 1.09421e6 1.34489
903903 2306.13i 0.00282819i
904904 403.451i 0.000493689i
905905 −700568. + 1.33124e6i −0.855369 + 1.62539i
906906 940238. 1.14546
907907 281974. 0.342763 0.171381 0.985205i 0.445177π-0.445177\pi
0.171381 + 0.985205i 0.445177π0.445177\pi
908908 −409387. −0.496549
909909 104727. 0.126745
910910 −131228. 69059.2i −0.158469 0.0833947i
911911 1.44668e6i 1.74315i −0.490260 0.871576i 0.663098π-0.663098\pi
0.490260 0.871576i 0.336902π-0.336902\pi
912912 −639053. −0.768329
913913 1.56222e6i 1.87413i
914914 910396.i 1.08978i
915915 481348. + 253311.i 0.574932 + 0.302560i
916916 591290.i 0.704709i
917917 198761. 0.236370
918918 −510093. −0.605291
919919 86628.3i 0.102572i −0.998684 0.0512860i 0.983668π-0.983668\pi
0.998684 0.0512860i 0.0163320π-0.0163320\pi
920920 305285. + 387371.i 0.360686 + 0.457668i
921921 −548169. −0.646243
922922 194997.i 0.229385i
923923 110983.i 0.130272i
924924 332689. 0.389667
925925 −166882. 242919.i −0.195041 0.283908i
926926 777780. 0.907058
927927 −213462. −0.248405
928928 778981.i 0.904547i
929929 169901. 0.196863 0.0984316 0.995144i 0.468617π-0.468617\pi
0.0984316 + 0.995144i 0.468617π0.468617\pi
930930 −272938. 143635.i −0.315572 0.166071i
931931 44603.6i 0.0514601i
932932 548753.i 0.631749i
933933 681796.i 0.783233i
934934 1.96487e6i 2.25237i
935935 185135. 351797.i 0.211770 0.402410i
936936 −32422.9 −0.0370084
937937 1.28498e6 1.46358 0.731792 0.681529i 0.238684π-0.238684\pi
0.731792 + 0.681529i 0.238684π0.238684\pi
938938 1.42917e6i 1.62435i
939939 67168.3i 0.0761786i
940940 −250183. + 475403.i −0.283140 + 0.538030i
941941 942315.i 1.06418i 0.846687 + 0.532092i 0.178594π0.178594\pi
−0.846687 + 0.532092i 0.821406π0.821406\pi
942942 1.56679e6 1.76567
943943 391979. 880283.i 0.440798 0.989918i
944944 1.29638e6 1.45475
945945 434938. 826479.i 0.487039 0.925483i
946946 4278.26 0.00478062
947947 819939.i 0.914285i −0.889394 0.457142i 0.848873π-0.848873\pi
0.889394 0.457142i 0.151127π-0.151127\pi
948948 −509735. −0.567189
949949 81369.7 0.0903505
950950 512223. + 745610.i 0.567560 + 0.826160i
951951 −715837. −0.791504
952952 231524.i 0.255459i
953953 143611. 0.158125 0.0790625 0.996870i 0.474807π-0.474807\pi
0.0790625 + 0.996870i 0.474807π0.474807\pi
954954 484845.i 0.532729i
955955 −565423. + 1.07443e6i −0.619964 + 1.17807i
956956 246067. 0.269239
957957 655998. 0.716273
958958 678192. 0.738961
959959 393573. 0.427945
960960 −19546.4 + 37142.6i −0.0212092 + 0.0403023i
961961 −790063. −0.855491
962962 58985.4 0.0637374
963963 64057.1 0.0690740
964964 515577.i 0.554803i
965965 −266050. + 505555.i −0.285699 + 0.542893i
966966 344596. 773874.i 0.369280 0.829309i
967967 1.68061e6i 1.79728i −0.438691 0.898638i 0.644558π-0.644558\pi
0.438691 0.898638i 0.355442π-0.355442\pi
968968 4158.17i 0.00443764i
969969 261624. 0.278632
970970 −893676. + 1.69819e6i −0.949810 + 1.80485i
971971 1.16288e6i 1.23338i 0.787207 + 0.616689i 0.211526π0.211526\pi
−0.787207 + 0.616689i 0.788474π0.788474\pi
972972 389901.i 0.412688i
973973 1.15801e6 1.22317
974974 −1.64784e6 −1.73699
975975 −61121.4 88970.5i −0.0642960 0.0935915i
976976 1.01895e6i 1.06967i
977977 997.036 0.00104453 0.000522266 1.00000i 0.499834π-0.499834\pi
0.000522266 1.00000i 0.499834π0.499834\pi
978978 996909.i 1.04226i
979979 666895. 0.695812
980980 28523.4 + 15010.6i 0.0296995 + 0.0156295i
981981 129407.i 0.134468i
982982 359866.i 0.373180i
983983 597261. 0.618097 0.309049 0.951046i 0.399989π-0.399989\pi
0.309049 + 0.951046i 0.399989π0.399989\pi
984984 463872. 0.479080
985985 165120. 313765.i 0.170187 0.323394i
986986 512129.i 0.526775i
987987 −822521. −0.844331
988988 −62615.8 −0.0641461
989989 1532.60 3441.82i 0.00156688 0.00351881i
990990 −456805. 240395.i −0.466080 0.245276i
991991 307119. 0.312722 0.156361 0.987700i 0.450024π-0.450024\pi
0.156361 + 0.987700i 0.450024π0.450024\pi
992992 359788.i 0.365614i
993993 259703.i 0.263377i
994994 1.02905e6i 1.04151i
995995 −518686. + 985619.i −0.523912 + 0.995550i
996996 742950.i 0.748930i
997997 1.32569e6i 1.33368i −0.745202 0.666839i 0.767647π-0.767647\pi
0.745202 0.666839i 0.232353π-0.232353\pi
998998 320475.i 0.321761i
999999 371493.i 0.372237i
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 115.5.c.c.114.12 yes 44
5.4 even 2 inner 115.5.c.c.114.33 yes 44
23.22 odd 2 inner 115.5.c.c.114.11 44
115.114 odd 2 inner 115.5.c.c.114.34 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.5.c.c.114.11 44 23.22 odd 2 inner
115.5.c.c.114.12 yes 44 1.1 even 1 trivial
115.5.c.c.114.33 yes 44 5.4 even 2 inner
115.5.c.c.114.34 yes 44 115.114 odd 2 inner