Properties

Label 504.2.cc.a.293.1
Level 504504
Weight 22
Character 504.293
Analytic conductor 4.0244.024
Analytic rank 00
Dimension 1616
CM discriminant -56
Inner twists 88

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(293,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 504=23327 504 = 2^{3} \cdot 3^{2} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 504.cc (of order 66, degree 22, 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: 4.024460261874.02446026187
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x16+)\mathbb{Q}[x]/(x^{16} + \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x16+10x12+19x8+810x4+6561 x^{16} + 10x^{12} + 19x^{8} + 810x^{4} + 6561 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 263 2^{6}\cdot 3
Twist minimal: yes
Sato-Tate group: U(1)[D6]\mathrm{U}(1)[D_{6}]

Embedding invariants

Embedding label 293.1
Root 0.475594+1.66548i-0.475594 + 1.66548i of defining polynomial
Character χ\chi == 504.293
Dual form 504.2.cc.a.461.1

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
f(q)f(q) == q+(1.22474+0.707107i)q2+(1.680140.420861i)q3+(1.000001.73205i)q4+(3.32070+1.91721i)q5+(2.355340.672592i)q6+(1.32288+2.29129i)q7+2.82843iq8+(2.64575+1.41421i)q95.42268q10+(2.40909+2.48923i)q12+(2.275333.94099i)q13+(3.240371.87083i)q14+(4.772374.61873i)q15+(2.000003.46410i)q16+(4.24037+0.138778i)q187.01226q19+(6.641403.83441i)q20+(1.258304.40644i)q21+(7.74037+4.46890i)q23+(1.190384.75216i)q24+(4.85136+8.40280i)q25+6.43560iq26+(3.850053.48957i)q27+5.29150q28+(9.11087+2.28220i)q30+(4.89898+2.82843i)q32+10.1449iq35+(5.095243.16836i)q36+(8.588234.95842i)q38+(5.48148+5.66382i)q39+(5.42268+9.39235i)q40+(4.65692+4.50700i)q42+(6.07440+9.76863i)q4512.6400q46+(1.90238+6.66190i)q48+(3.50000+6.06218i)q49+(11.88346.86086i)q50+(4.550667.88197i)q52+(7.18283+1.55144i)q54+(6.48074+3.74166i)q56+(11.7816+2.95119i)q57+(4.58621+2.64785i)q59+(12.7722+3.64725i)q60+(3.66740+6.35212i)q61+(0.259630+7.93301i)q638.00000q64+(15.11148.72455i)q65+(11.124110.7660i)q69+(7.1735312.4249i)q70+10.9193iq71+(4.00000+7.48331i)q72+(4.6145616.1596i)q75+(7.01226+12.1456i)q76+(2.7085010.8127i)q78+(8.6713415.0192i)q7915.3376iq80+(5.00000+7.48331i)q81+(15.73309.08345i)q83+(8.890472.22699i)q84+(14.34717.66882i)q90+12.0399q91+(15.48078.93781i)q92+(23.285613.4439i)q95+(7.040606.81395i)q969.89949iq98+O(q100)q+(-1.22474 + 0.707107i) q^{2} +(-1.68014 - 0.420861i) q^{3} +(1.00000 - 1.73205i) q^{4} +(3.32070 + 1.91721i) q^{5} +(2.35534 - 0.672592i) q^{6} +(1.32288 + 2.29129i) q^{7} +2.82843i q^{8} +(2.64575 + 1.41421i) q^{9} -5.42268 q^{10} +(-2.40909 + 2.48923i) q^{12} +(2.27533 - 3.94099i) q^{13} +(-3.24037 - 1.87083i) q^{14} +(-4.77237 - 4.61873i) q^{15} +(-2.00000 - 3.46410i) q^{16} +(-4.24037 + 0.138778i) q^{18} -7.01226 q^{19} +(6.64140 - 3.83441i) q^{20} +(-1.25830 - 4.40644i) q^{21} +(7.74037 + 4.46890i) q^{23} +(1.19038 - 4.75216i) q^{24} +(4.85136 + 8.40280i) q^{25} +6.43560i q^{26} +(-3.85005 - 3.48957i) q^{27} +5.29150 q^{28} +(9.11087 + 2.28220i) q^{30} +(4.89898 + 2.82843i) q^{32} +10.1449i q^{35} +(5.09524 - 3.16836i) q^{36} +(8.58823 - 4.95842i) q^{38} +(-5.48148 + 5.66382i) q^{39} +(-5.42268 + 9.39235i) q^{40} +(4.65692 + 4.50700i) q^{42} +(6.07440 + 9.76863i) q^{45} -12.6400 q^{46} +(1.90238 + 6.66190i) q^{48} +(-3.50000 + 6.06218i) q^{49} +(-11.8834 - 6.86086i) q^{50} +(-4.55066 - 7.88197i) q^{52} +(7.18283 + 1.55144i) q^{54} +(-6.48074 + 3.74166i) q^{56} +(11.7816 + 2.95119i) q^{57} +(4.58621 + 2.64785i) q^{59} +(-12.7722 + 3.64725i) q^{60} +(3.66740 + 6.35212i) q^{61} +(0.259630 + 7.93301i) q^{63} -8.00000 q^{64} +(15.1114 - 8.72455i) q^{65} +(-11.1241 - 10.7660i) q^{69} +(-7.17353 - 12.4249i) q^{70} +10.9193i q^{71} +(-4.00000 + 7.48331i) q^{72} +(-4.61456 - 16.1596i) q^{75} +(-7.01226 + 12.1456i) q^{76} +(2.70850 - 10.8127i) q^{78} +(-8.67134 - 15.0192i) q^{79} -15.3376i q^{80} +(5.00000 + 7.48331i) q^{81} +(15.7330 - 9.08345i) q^{83} +(-8.89047 - 2.22699i) q^{84} +(-14.3471 - 7.66882i) q^{90} +12.0399 q^{91} +(15.4807 - 8.93781i) q^{92} +(-23.2856 - 13.4439i) q^{95} +(-7.04060 - 6.81395i) q^{96} -9.89949i q^{98} +O(q^{100})
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q+16q416q1532q1616q18+72q23+40q25+32q3040q3956q49144q50+8q5716q60+56q63128q6472q6564q72+128q78++144q92+O(q100) 16 q + 16 q^{4} - 16 q^{15} - 32 q^{16} - 16 q^{18} + 72 q^{23} + 40 q^{25} + 32 q^{30} - 40 q^{39} - 56 q^{49} - 144 q^{50} + 8 q^{57} - 16 q^{60} + 56 q^{63} - 128 q^{64} - 72 q^{65} - 64 q^{72} + 128 q^{78}+ \cdots + 144 q^{92}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/504Z)×\left(\mathbb{Z}/504\mathbb{Z}\right)^\times.

nn 7373 127127 253253 281281
χ(n)\chi(n) 1-1 11 1-1 e(56)e\left(\frac{5}{6}\right)

Coefficient data

For each nn we display the coefficients of the qq-expansion ana_n, the Satake parameters αp\alpha_p, and the Satake angles θp=Arg(αp)\theta_p = \textrm{Arg}(\alpha_p).



Display apa_p with pp up to: 50 250 1000 (See ana_n instead) (See ana_n instead) (See ana_n instead) Display ana_n with nn up to: 50 250 1000 (See only apa_p) (See only apa_p) (See only apa_p)
Significant digits:
nn ana_n an/n(k1)/2a_n / n^{(k-1)/2} αn \alpha_n θn \theta_n
pp apa_p ap/p(k1)/2a_p / p^{(k-1)/2} αp \alpha_p θp \theta_p
22 −1.22474 + 0.707107i −0.866025 + 0.500000i
33 −1.68014 0.420861i −0.970030 0.242984i
44 1.00000 1.73205i 0.500000 0.866025i
55 3.32070 + 1.91721i 1.48506 + 0.857401i 0.999856 0.0169992i 0.00541128π-0.00541128\pi
0.485206 + 0.874400i 0.338745π0.338745\pi
66 2.35534 0.672592i 0.961563 0.274584i
77 1.32288 + 2.29129i 0.500000 + 0.866025i
88 2.82843i 1.00000i
99 2.64575 + 1.41421i 0.881917 + 0.471405i
1010 −5.42268 −1.71480
1111 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
1212 −2.40909 + 2.48923i −0.695446 + 0.718579i
1313 2.27533 3.94099i 0.631063 1.09303i −0.356272 0.934382i 0.615952π-0.615952\pi
0.987335 0.158651i 0.0507143π-0.0507143\pi
1414 −3.24037 1.87083i −0.866025 0.500000i
1515 −4.77237 4.61873i −1.23222 1.19255i
1616 −2.00000 3.46410i −0.500000 0.866025i
1717 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
1818 −4.24037 + 0.138778i −0.999465 + 0.0327103i
1919 −7.01226 −1.60872 −0.804361 0.594140i 0.797492π-0.797492\pi
−0.804361 + 0.594140i 0.797492π0.797492\pi
2020 6.64140 3.83441i 1.48506 0.857401i
2121 −1.25830 4.40644i −0.274584 0.961563i
2222 0 0
2323 7.74037 + 4.46890i 1.61398 + 0.931831i 0.988436 + 0.151642i 0.0484560π0.0484560\pi
0.625543 + 0.780189i 0.284877π0.284877\pi
2424 1.19038 4.75216i 0.242984 0.970030i
2525 4.85136 + 8.40280i 0.970272 + 1.68056i
2626 6.43560i 1.26213i
2727 −3.85005 3.48957i −0.740942 0.671569i
2828 5.29150 1.00000
2929 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
3030 9.11087 + 2.28220i 1.66341 + 0.416670i
3131 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
3232 4.89898 + 2.82843i 0.866025 + 0.500000i
3333 0 0
3434 0 0
3535 10.1449i 1.71480i
3636 5.09524 3.16836i 0.849207 0.528060i
3737 0 0 1.00000 00
−1.00000 π\pi
3838 8.58823 4.95842i 1.39319 0.804361i
3939 −5.48148 + 5.66382i −0.877740 + 0.906936i
4040 −5.42268 + 9.39235i −0.857401 + 1.48506i
4141 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
4242 4.65692 + 4.50700i 0.718579 + 0.695446i
4343 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
4444 0 0
4545 6.07440 + 9.76863i 0.905519 + 1.45622i
4646 −12.6400 −1.86366
4747 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
4848 1.90238 + 6.66190i 0.274584 + 0.961563i
4949 −3.50000 + 6.06218i −0.500000 + 0.866025i
5050 −11.8834 6.86086i −1.68056 0.970272i
5151 0 0
5252 −4.55066 7.88197i −0.631063 1.09303i
5353 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
5454 7.18283 + 1.55144i 0.977459 + 0.211124i
5555 0 0
5656 −6.48074 + 3.74166i −0.866025 + 0.500000i
5757 11.7816 + 2.95119i 1.56051 + 0.390895i
5858 0 0
5959 4.58621 + 2.64785i 0.597074 + 0.344721i 0.767890 0.640582i 0.221307π-0.221307\pi
−0.170816 + 0.985303i 0.554640π0.554640\pi
6060 −12.7722 + 3.64725i −1.64889 + 0.470858i
6161 3.66740 + 6.35212i 0.469562 + 0.813306i 0.999394 0.0347968i 0.0110784π-0.0110784\pi
−0.529832 + 0.848103i 0.677745π0.677745\pi
6262 0 0
6363 0.259630 + 7.93301i 0.0327103 + 0.999465i
6464 −8.00000 −1.00000
6565 15.1114 8.72455i 1.87433 1.08215i
6666 0 0
6767 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
6868 0 0
6969 −11.1241 10.7660i −1.33919 1.29608i
7070 −7.17353 12.4249i −0.857401 1.48506i
7171 10.9193i 1.29588i 0.761690 + 0.647941i 0.224370π0.224370\pi
−0.761690 + 0.647941i 0.775630π0.775630\pi
7272 −4.00000 + 7.48331i −0.471405 + 0.881917i
7373 0 0 1.00000 00
−1.00000 π\pi
7474 0 0
7575 −4.61456 16.1596i −0.532843 1.86595i
7676 −7.01226 + 12.1456i −0.804361 + 1.39319i
7777 0 0
7878 2.70850 10.8127i 0.306677 1.22430i
7979 −8.67134 15.0192i −0.975603 1.68979i −0.677932 0.735124i 0.737124π-0.737124\pi
−0.297670 0.954669i 0.596210π-0.596210\pi
8080 15.3376i 1.71480i
8181 5.00000 + 7.48331i 0.555556 + 0.831479i
8282 0 0
8383 15.7330 9.08345i 1.72692 0.997039i 0.825010 0.565118i 0.191170π-0.191170\pi
0.901912 0.431920i 0.142164π-0.142164\pi
8484 −8.89047 2.22699i −0.970030 0.242984i
8585 0 0
8686 0 0
8787 0 0
8888 0 0
8989 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
9090 −14.3471 7.66882i −1.51231 0.808365i
9191 12.0399 1.26213
9292 15.4807 8.93781i 1.61398 0.931831i
9393 0 0
9494 0 0
9595 −23.2856 13.4439i −2.38905 1.37932i
9696 −7.04060 6.81395i −0.718579 0.695446i
9797 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
9898 9.89949i 1.00000i
9999 0 0
100100 19.4054 1.94054
101101 −17.3740 + 10.0309i −1.72877 + 0.998109i −0.833650 + 0.552294i 0.813753π0.813753\pi
−0.895125 + 0.445815i 0.852914π0.852914\pi
102102 0 0
103103 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
104104 11.1468 + 6.43560i 1.09303 + 0.631063i
105105 4.26960 17.0449i 0.416670 1.66341i
106106 0 0
107107 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
108108 −9.89417 + 3.17891i −0.952067 + 0.305890i
109109 0 0 1.00000 00
−1.00000 π\pi
110110 0 0
111111 0 0
112112 5.29150 9.16515i 0.500000 0.866025i
113113 −5.78216 3.33833i −0.543940 0.314044i 0.202735 0.979234i 0.435017π-0.435017\pi
−0.746674 + 0.665190i 0.768350π0.768350\pi
114114 −16.5162 + 4.71639i −1.54689 + 0.441730i
115115 17.1356 + 29.6798i 1.59791 + 2.76765i
116116 0 0
117117 11.5934 7.20907i 1.07181 0.666479i
118118 −7.48925 −0.689442
119119 0 0
120120 13.0637 13.4983i 1.19255 1.23222i
121121 5.50000 9.52628i 0.500000 0.866025i
122122 −8.98326 5.18648i −0.813306 0.469562i
123123 0 0
124124 0 0
125125 18.0322i 1.61285i
126126 −5.92746 9.53232i −0.528060 0.849207i
127127 18.4422 1.63648 0.818241 0.574875i 0.194949π-0.194949\pi
0.818241 + 0.574875i 0.194949π0.194949\pi
128128 9.79796 5.65685i 0.866025 0.500000i
129129 0 0
130130 −12.3384 + 21.3707i −1.08215 + 1.87433i
131131 −8.39072 4.84438i −0.733100 0.423256i 0.0864550 0.996256i 0.472446π-0.472446\pi
−0.819555 + 0.573000i 0.805779π0.805779\pi
132132 0 0
133133 −9.27635 16.0671i −0.804361 1.39319i
134134 0 0
135135 −6.09462 18.9692i −0.524541 1.63261i
136136 0 0
137137 −12.9615 + 7.48331i −1.10737 + 0.639343i −0.938148 0.346235i 0.887460π-0.887460\pi
−0.169226 + 0.985577i 0.554127π0.554127\pi
138138 21.2369 + 5.31968i 1.80781 + 0.452841i
139139 −1.14233 + 1.97858i −0.0968913 + 0.167821i −0.910396 0.413737i 0.864223π-0.864223\pi
0.813505 + 0.581558i 0.197557π0.197557\pi
140140 17.5715 + 10.1449i 1.48506 + 0.857401i
141141 0 0
142142 −7.72111 13.3734i −0.647941 1.12227i
143143 0 0
144144 −0.392523 11.9936i −0.0327103 0.999465i
145145 0 0
146146 0 0
147147 8.43183 8.71230i 0.695446 0.718579i
148148 0 0
149149 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
150150 17.0782 + 16.5285i 1.39443 + 1.34954i
151151 −12.2211 21.1676i −0.994540 1.72259i −0.587646 0.809118i 0.699945π-0.699945\pi
−0.406894 0.913475i 0.633388π-0.633388\pi
152152 19.8337i 1.60872i
153153 0 0
154154 0 0
155155 0 0
156156 4.32853 + 15.1580i 0.346560 + 1.21361i
157157 3.34486 5.79347i 0.266949 0.462369i −0.701123 0.713040i 0.747318π-0.747318\pi
0.968072 + 0.250671i 0.0806511π0.0806511\pi
158158 21.2404 + 12.2631i 1.68979 + 0.975603i
159159 0 0
160160 10.8454 + 18.7847i 0.857401 + 1.48506i
161161 23.6472i 1.86366i
162162 −11.4152 5.62962i −0.896865 0.442305i
163163 0 0 1.00000 00
−1.00000 π\pi
164164 0 0
165165 0 0
166166 −12.8459 + 22.2498i −0.997039 + 1.72692i
167167 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
168168 12.4633 3.55902i 0.961563 0.274584i
169169 −3.85425 6.67575i −0.296481 0.513520i
170170 0 0
171171 −18.5527 9.91683i −1.41876 0.758359i
172172 0 0
173173 −21.3064 + 12.3013i −1.61990 + 0.935247i −0.632950 + 0.774193i 0.718156π0.718156\pi
−0.986945 + 0.161055i 0.948510π0.948510\pi
174174 0 0
175175 −12.8355 + 22.2317i −0.970272 + 1.68056i
176176 0 0
177177 −6.59111 6.37892i −0.495418 0.479469i
178178 0 0
179179 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
180180 22.9942 0.752548i 1.71388 0.0560916i
181181 3.13802 0.233247 0.116623 0.993176i 0.462793π-0.462793\pi
0.116623 + 0.993176i 0.462793π0.462793\pi
182182 −14.7458 + 8.51350i −1.09303 + 0.631063i
183183 −3.48839 12.2159i −0.257869 0.903027i
184184 −12.6400 + 21.8931i −0.931831 + 1.61398i
185185 0 0
186186 0 0
187187 0 0
188188 0 0
189189 2.90248 13.4378i 0.211124 0.977459i
190190 38.0252 2.75864
191191 2.10792 1.21701i 0.152524 0.0880597i −0.421796 0.906691i 0.638600π-0.638600\pi
0.574320 + 0.818631i 0.305267π0.305267\pi
192192 13.4411 + 3.36689i 0.970030 + 0.242984i
193193 10.2886 17.8204i 0.740591 1.28274i −0.211636 0.977349i 0.567879π-0.567879\pi
0.952227 0.305392i 0.0987875π-0.0987875\pi
194194 0 0
195195 −29.0611 + 8.29869i −2.08111 + 0.594282i
196196 7.00000 + 12.1244i 0.500000 + 0.866025i
197197 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
198198 0 0
199199 0 0 1.00000 00
−1.00000 π\pi
200200 −23.7667 + 13.7217i −1.68056 + 0.970272i
201201 0 0
202202 14.1858 24.5705i 0.998109 1.72877i
203203 0 0
204204 0 0
205205 0 0
206206 0 0
207207 14.1591 + 22.7701i 0.984126 + 1.58263i
208208 −18.2026 −1.26213
209209 0 0
210210 6.82338 + 23.8947i 0.470858 + 1.64889i
211211 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
212212 0 0
213213 4.59551 18.3460i 0.314879 1.25705i
214214 0 0
215215 0 0
216216 9.87000 10.8896i 0.671569 0.740942i
217217 0 0
218218 0 0
219219 0 0
220220 0 0
221221 0 0
222222 0 0
223223 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
224224 14.9666i 1.00000i
225225 0.952135 + 29.0926i 0.0634757 + 1.93951i
226226 9.44222 0.628087
227227 −12.7602 + 7.36712i −0.846926 + 0.488973i −0.859612 0.510947i 0.829295π-0.829295\pi
0.0126866 + 0.999920i 0.495962π0.495962\pi
228228 16.8932 17.4551i 1.11878 1.15599i
229229 14.6990 25.4595i 0.971339 1.68241i 0.279817 0.960053i 0.409726π-0.409726\pi
0.691522 0.722355i 0.256940π-0.256940\pi
230230 −41.9735 24.2334i −2.76765 1.59791i
231231 0 0
232232 0 0
233233 20.1628i 1.32091i −0.750867 0.660454i 0.770364π-0.770364\pi
0.750867 0.660454i 0.229636π-0.229636\pi
234234 −9.10132 + 17.0270i −0.594972 + 1.11309i
235235 0 0
236236 9.17242 5.29570i 0.597074 0.344721i
237237 8.24808 + 28.8838i 0.535770 + 1.87621i
238238 0 0
239239 −19.2596 11.1196i −1.24580 0.719264i −0.275533 0.961292i 0.588854π-0.588854\pi
−0.970269 + 0.242028i 0.922188π0.922188\pi
240240 −6.45503 + 25.7694i −0.416670 + 1.66341i
241241 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
242242 15.5563i 1.00000i
243243 −5.25127 14.6773i −0.336869 0.941551i
244244 14.6696 0.939125
245245 −23.2449 + 13.4204i −1.48506 + 0.857401i
246246 0 0
247247 −15.9552 + 27.6352i −1.01521 + 1.75839i
248248 0 0
249249 −30.2566 + 8.64007i −1.91743 + 0.547542i
250250 −12.7507 22.0848i −0.806423 1.39677i
251251 7.06541i 0.445965i 0.974822 + 0.222982i 0.0715793π0.0715793\pi
−0.974822 + 0.222982i 0.928421π0.928421\pi
252252 14.0000 + 7.48331i 0.881917 + 0.471405i
253253 0 0
254254 −22.5870 + 13.0406i −1.41724 + 0.818241i
255255 0 0
256256 −8.00000 + 13.8564i −0.500000 + 0.866025i
257257 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
258258 0 0
259259 0 0
260260 34.8982i 2.16429i
261261 0 0
262262 13.7020 0.846511
263263 24.1533 13.9449i 1.48936 0.859881i 0.489432 0.872041i 0.337204π-0.337204\pi
0.999926 + 0.0121601i 0.00387079π0.00387079\pi
264264 0 0
265265 0 0
266266 22.7223 + 13.1187i 1.39319 + 0.804361i
267267 0 0
268268 0 0
269269 20.5179i 1.25100i −0.780225 0.625498i 0.784896π-0.784896\pi
0.780225 0.625498i 0.215104π-0.215104\pi
270270 20.8776 + 18.9228i 1.27057 + 1.15161i
271271 0 0 1.00000 00
−1.00000 π\pi
272272 0 0
273273 −20.2288 5.06713i −1.22430 0.306677i
274274 10.5830 18.3303i 0.639343 1.10737i
275275 0 0
276276 −29.7714 + 8.50154i −1.79203 + 0.511733i
277277 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
278278 3.23100i 0.193783i
279279 0 0
280280 −28.6941 −1.71480
281281 28.9807 16.7320i 1.72885 0.998150i 0.834021 0.551733i 0.186033π-0.186033\pi
0.894825 0.446417i 0.147300π-0.147300\pi
282282 0 0
283283 3.82867 6.63145i 0.227591 0.394199i −0.729503 0.683978i 0.760249π-0.760249\pi
0.957094 + 0.289779i 0.0935819π0.0935819\pi
284284 18.9128 + 10.9193i 1.12227 + 0.647941i
285285 33.4651 + 32.3877i 1.98230 + 1.91848i
286286 0 0
287287 0 0
288288 8.96148 + 14.4115i 0.528060 + 0.849207i
289289 −17.0000 −1.00000
290290 0 0
291291 0 0
292292 0 0
293293 16.9789 + 9.80280i 0.991921 + 0.572686i 0.905848 0.423603i 0.139235π-0.139235\pi
0.0860728 + 0.996289i 0.472568π0.472568\pi
294294 −4.16632 + 16.6326i −0.242984 + 0.970030i
295295 10.1530 + 17.5854i 0.591128 + 1.02386i
296296 0 0
297297 0 0
298298 0 0
299299 35.2238 20.3365i 2.03704 1.17609i
300300 −32.6039 8.16700i −1.88239 0.471522i
301301 0 0
302302 29.9355 + 17.2833i 1.72259 + 0.994540i
303303 33.4123 9.54124i 1.91949 0.548130i
304304 14.0245 + 24.2912i 0.804361 + 1.39319i
305305 28.1246i 1.61041i
306306 0 0
307307 34.8207 1.98732 0.993662 0.112409i 0.0358567π-0.0358567\pi
0.993662 + 0.112409i 0.0358567π0.0358567\pi
308308 0 0
309309 0 0
310310 0 0
311311 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
312312 −16.0197 15.5040i −0.906936 0.877740i
313313 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
314314 9.46070i 0.533898i
315315 −14.3471 + 26.8409i −0.808365 + 1.51231i
316316 −34.6854 −1.95121
317317 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
318318 0 0
319319 0 0
320320 −26.5656 15.3376i −1.48506 0.857401i
321321 0 0
322322 −16.7211 28.9618i −0.931831 1.61398i
323323 0 0
324324 17.9615 1.17694i 0.997860 0.0653855i
325325 44.1538 2.44921
326326 0 0
327327 0 0
328328 0 0
329329 0 0
330330 0 0
331331 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
332332 36.3338i 1.99408i
333333 0 0
334334 0 0
335335 0 0
336336 −12.7477 + 13.1718i −0.695446 + 0.718579i
337337 −13.2288 + 22.9129i −0.720616 + 1.24814i 0.240137 + 0.970739i 0.422808π0.422808\pi
−0.960753 + 0.277405i 0.910526π0.910526\pi
338338 9.44094 + 5.45073i 0.513520 + 0.296481i
339339 8.30987 + 8.04235i 0.451330 + 0.436801i
340340 0 0
341341 0 0
342342 29.7346 0.973147i 1.60786 0.0526217i
343343 −18.5203 −1.00000
344344 0 0
345345 −16.2992 57.0779i −0.877520 3.07297i
346346 17.3966 30.1318i 0.935247 1.61990i
347347 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
348348 0 0
349349 −14.3152 24.7947i −0.766277 1.32723i −0.939569 0.342361i 0.888774π-0.888774\pi
0.173291 0.984871i 0.444560π-0.444560\pi
350350 36.3042i 1.94054i
351351 −22.5125 + 7.23306i −1.20163 + 0.386072i
352352 0 0
353353 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
354354 12.5830 + 3.15194i 0.668779 + 0.167524i
355355 −20.9345 + 36.2597i −1.11109 + 1.92447i
356356 0 0
357357 0 0
358358 0 0
359359 13.5121i 0.713143i 0.934268 + 0.356572i 0.116054π0.116054\pi
−0.934268 + 0.356572i 0.883946π0.883946\pi
360360 −27.6299 + 17.1810i −1.45622 + 0.905519i
361361 30.1718 1.58799
362362 −3.84327 + 2.21891i −0.201998 + 0.116623i
363363 −13.2500 + 13.6908i −0.695446 + 0.718579i
364364 12.0399 20.8537i 0.631063 1.09303i
365365 0 0
366366 12.9103 + 12.4947i 0.674835 + 0.653110i
367367 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
368368 35.7512i 1.86366i
369369 0 0
370370 0 0
371371 0 0
372372 0 0
373373 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
374374 0 0
375375 7.58904 30.2966i 0.391896 1.56451i
376376 0 0
377377 0 0
378378 5.94719 + 18.5103i 0.305890 + 0.952067i
379379 0 0 1.00000 00
−1.00000 π\pi
380380 −46.5712 + 26.8879i −2.38905 + 1.37932i
381381 −30.9855 7.76162i −1.58744 0.397640i
382382 −1.72111 + 2.98105i −0.0880597 + 0.152524i
383383 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
384384 −18.8427 + 5.38073i −0.961563 + 0.274584i
385385 0 0
386386 29.1006i 1.48118i
387387 0 0
388388 0 0
389389 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
390390 29.7243 30.7131i 1.50515 1.55522i
391391 0 0
392392 −17.1464 9.89949i −0.866025 0.500000i
393393 12.0588 + 11.6706i 0.608285 + 0.588703i
394394 0 0
395395 66.4990i 3.34593i
396396 0 0
397397 −37.7318 −1.89370 −0.946852 0.321668i 0.895756π-0.895756\pi
−0.946852 + 0.321668i 0.895756π0.895756\pi
398398 0 0
399399 8.82355 + 30.8991i 0.441730 + 1.54689i
400400 19.4054 33.6112i 0.970272 1.68056i
401401 −27.8276 16.0663i −1.38964 0.802310i −0.396368 0.918092i 0.629729π-0.629729\pi
−0.993275 + 0.115782i 0.963063π0.963063\pi
402402 0 0
403403 0 0
404404 40.1235i 1.99622i
405405 2.25644 + 34.4359i 0.112123 + 1.71113i
406406 0 0
407407 0 0
408408 0 0
409409 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
410410 0 0
411411 24.9266 7.11804i 1.22954 0.351107i
412412 0 0
413413 14.0111i 0.689442i
414414 −33.4422 17.8756i −1.64360 0.878539i
415415 69.6594 3.41945
416416 22.2936 12.8712i 1.09303 0.631063i
417417 2.75198 2.84352i 0.134765 0.139248i
418418 0 0
419419 −7.16397 4.13612i −0.349983 0.202063i 0.314695 0.949193i 0.398098π-0.398098\pi
−0.664678 + 0.747130i 0.731431π0.731431\pi
420420 −25.2530 24.4400i −1.23222 1.19255i
421421 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
422422 0 0
423423 0 0
424424 0 0
425425 0 0
426426 7.34423 + 25.7187i 0.355829 + 1.24607i
427427 −9.70302 + 16.8061i −0.469562 + 0.813306i
428428 0 0
429429 0 0
430430 0 0
431431 37.4166i 1.80229i −0.433515 0.901146i 0.642727π-0.642727\pi
0.433515 0.901146i 0.357273π-0.357273\pi
432432 −4.38814 + 20.3161i −0.211124 + 0.977459i
433433 0 0 1.00000 00
−1.00000 π\pi
434434 0 0
435435 0 0
436436 0 0
437437 −54.2775 31.3371i −2.59644 1.49906i
438438 0 0
439439 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
440440 0 0
441441 −17.8333 + 11.0893i −0.849207 + 0.528060i
442442 0 0
443443 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
444444 0 0
445445 0 0
446446 0 0
447447 0 0
448448 −10.5830 18.3303i −0.500000 0.866025i
449449 11.0141i 0.519789i 0.965637 + 0.259895i 0.0836878π0.0836878\pi
−0.965637 + 0.259895i 0.916312π0.916312\pi
450450 −21.7377 34.9577i −1.02472 1.64792i
451451 0 0
452452 −11.5643 + 6.67666i −0.543940 + 0.314044i
453453 11.6246 + 40.7079i 0.546170 + 1.91263i
454454 10.4187 18.0457i 0.488973 0.846926i
455455 39.9809 + 23.0830i 1.87433 + 1.08215i
456456 −8.34723 + 33.3234i −0.390895 + 1.56051i
457457 −19.6940 34.1111i −0.921249 1.59565i −0.797486 0.603338i 0.793837π-0.793837\pi
−0.123763 0.992312i 0.539496π-0.539496\pi
458458 41.5751i 1.94268i
459459 0 0
460460 68.5425 3.19581
461461 34.5504 19.9477i 1.60917 0.929057i 0.619619 0.784903i 0.287287π-0.287287\pi
0.989555 0.144154i 0.0460461π-0.0460461\pi
462462 0 0
463463 21.3113 36.9123i 0.990421 1.71546i 0.375628 0.926770i 0.377427π-0.377427\pi
0.614792 0.788689i 0.289240π-0.289240\pi
464464 0 0
465465 0 0
466466 14.2572 + 24.6943i 0.660454 + 1.14394i
467467 41.6295i 1.92638i 0.268814 + 0.963192i 0.413368π0.413368\pi
−0.268814 + 0.963192i 0.586632π0.586632\pi
468468 −0.893119 27.2893i −0.0412845 1.26145i
469469 0 0
470470 0 0
471471 −8.05809 + 8.32613i −0.371297 + 0.383648i
472472 −7.48925 + 12.9718i −0.344721 + 0.597074i
473473 0 0
474474 −30.5257 29.5431i −1.40209 1.35696i
475475 −34.0190 58.9226i −1.56090 2.70356i
476476 0 0
477477 0 0
478478 31.4508 1.43853
479479 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
480480 −10.3160 36.1254i −0.470858 1.64889i
481481 0 0
482482 0 0
483483 9.95220 39.7307i 0.452841 1.80781i
484484 −11.0000 19.0526i −0.500000 0.866025i
485485 0 0
486486 16.8099 + 14.2628i 0.762513 + 0.646973i
487487 0.442222 0.0200390 0.0100195 0.999950i 0.496811π-0.496811\pi
0.0100195 + 0.999950i 0.496811π0.496811\pi
488488 −17.9665 + 10.3730i −0.813306 + 0.469562i
489489 0 0
490490 18.9794 32.8732i 0.857401 1.48506i
491491 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
492492 0 0
493493 0 0
494494 45.1281i 2.03041i
495495 0 0
496496 0 0
497497 −25.0193 + 14.4449i −1.12227 + 0.647941i
498498 30.9471 31.9765i 1.38677 1.43290i
499499 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
500500 31.2326 + 18.0322i 1.39677 + 0.806423i
501501 0 0
502502 −4.99600 8.65333i −0.222982 0.386217i
503503 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
504504 −22.4379 + 0.734344i −0.999465 + 0.0327103i
505505 −76.9250 −3.42312
506506 0 0
507507 3.66612 + 12.8383i 0.162818 + 0.570170i
508508 18.4422 31.9429i 0.818241 1.41724i
509509 −32.4532 18.7369i −1.43846 0.830497i −0.440719 0.897645i 0.645277π-0.645277\pi
−0.997743 + 0.0671482i 0.978610π0.978610\pi
510510 0 0
511511 0 0
512512 22.6274i 1.00000i
513513 26.9975 + 24.4698i 1.19197 + 1.08037i
514514 0 0
515515 0 0
516516 0 0
517517 0 0
518518 0 0
519519 40.9749 11.7008i 1.79860 0.513609i
520520 24.6768 + 42.7414i 1.08215 + 1.87433i
521521 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
522522 0 0
523523 −45.6661 −1.99684 −0.998419 0.0562050i 0.982100π-0.982100\pi
−0.998419 + 0.0562050i 0.982100π0.982100\pi
524524 −16.7814 + 9.68876i −0.733100 + 0.423256i
525525 30.9219 31.9505i 1.34954 1.39443i
526526 −19.7211 + 34.1580i −0.859881 + 1.48936i
527527 0 0
528528 0 0
529529 28.4422 + 49.2634i 1.23662 + 2.14189i
530530 0 0
531531 8.38935 + 13.4914i 0.364067 + 0.585479i
532532 −37.1054 −1.60872
533533 0 0
534534 0 0
535535 0 0
536536 0 0
537537 0 0
538538 14.5083 + 25.1292i 0.625498 + 1.08339i
539539 0 0
540540 −38.9502 8.41297i −1.67615 0.362037i
541541 0 0 1.00000 00
−1.00000 π\pi
542542 0 0
543543 −5.27231 1.32067i −0.226257 0.0566754i
544544 0 0
545545 0 0
546546 28.3581 8.09794i 1.21361 0.346560i
547547 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
548548 29.9333i 1.27869i
549549 0.719769 + 21.9926i 0.0307190 + 0.938622i
550550 0 0
551551 0 0
552552 30.4509 31.4638i 1.29608 1.33919i
553553 22.9422 39.7371i 0.975603 1.68979i
554554 0 0
555555 0 0
556556 2.28466 + 3.95715i 0.0968913 + 0.167821i
557557 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
558558 0 0
559559 0 0
560560 35.1430 20.2898i 1.48506 0.857401i
561561 0 0
562562 −23.6627 + 40.9850i −0.998150 + 1.72885i
563563 13.8054 + 7.97053i 0.581827 + 0.335918i 0.761859 0.647743i 0.224287π-0.224287\pi
−0.180032 + 0.983661i 0.557620π0.557620\pi
564564 0 0
565565 −12.8005 22.1712i −0.538522 0.932748i
566566 10.8291i 0.455181i
567567 −10.5320 + 21.3559i −0.442305 + 0.896865i
568568 −30.8844 −1.29588
569569 −2.44949 + 1.41421i −0.102688 + 0.0592869i −0.550464 0.834859i 0.685549π-0.685549\pi
0.447777 + 0.894146i 0.352216π0.352216\pi
570570 −63.8878 16.0034i −2.67596 0.670307i
571571 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
572572 0 0
573573 −4.05380 + 1.15760i −0.169350 + 0.0483596i
574574 0 0
575575 86.7210i 3.61652i
576576 −21.1660 11.3137i −0.881917 0.471405i
577577 0 0 1.00000 00
−1.00000 π\pi
578578 20.8207 12.0208i 0.866025 0.500000i
579579 −24.7862 + 25.6107i −1.03008 + 1.06434i
580580 0 0
581581 41.6256 + 24.0326i 1.72692 + 0.997039i
582582 0 0
583583 0 0
584584 0 0
585585 52.3193 1.71229i 2.16314 0.0707947i
586586 −27.7265 −1.14537
587587 7.99569 4.61631i 0.330017 0.190536i −0.325831 0.945428i 0.605644π-0.605644\pi
0.655849 + 0.754892i 0.272311π0.272311\pi
588588 −6.65832 23.3167i −0.274584 0.961563i
589589 0 0
590590 −24.8696 14.3584i −1.02386 0.591128i
591591 0 0
592592 0 0
593593 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
594594 0 0
595595 0 0
596596 0 0
597597 0 0
598598 −28.7601 + 49.8140i −1.17609 + 2.03704i
599599 9.79796 + 5.65685i 0.400334 + 0.231133i 0.686628 0.727009i 0.259090π-0.259090\pi
−0.286294 + 0.958142i 0.592423π0.592423\pi
600600 45.7064 13.0519i 1.86595 0.532843i
601601 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
602602 0 0
603603 0 0
604604 −48.8844 −1.98908
605605 36.5277 21.0893i 1.48506 0.857401i
606606 −34.1749 + 35.3117i −1.38826 + 1.43444i
607607 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
608608 −34.3529 19.8337i −1.39319 0.804361i
609609 0 0
610610 −19.8871 34.4455i −0.805206 1.39466i
611611 0 0
612612 0 0
613613 0 0 1.00000 00
−1.00000 π\pi
614614 −42.6465 + 24.6220i −1.72107 + 0.993662i
615615 0 0
616616 0 0
617617 −12.2474 7.07107i −0.493064 0.284670i 0.232781 0.972529i 0.425218π-0.425218\pi
−0.725845 + 0.687859i 0.758551π0.758551\pi
618618 0 0
619619 24.8654 + 43.0682i 0.999427 + 1.73106i 0.529034 + 0.848601i 0.322554π0.322554\pi
0.470393 + 0.882457i 0.344112π0.344112\pi
620620 0 0
621621 −14.2062 44.2161i −0.570076 1.77433i
622622 0 0
623623 0 0
624624 30.5830 + 7.66079i 1.22430 + 0.306677i
625625 −10.3146 + 17.8654i −0.412583 + 0.714615i
626626 0 0
627627 0 0
628628 −6.68972 11.5869i −0.266949 0.462369i
629629 0 0
630630 −1.40789 43.0181i −0.0560916 1.71388i
631631 −10.8736 −0.432872 −0.216436 0.976297i 0.569443π-0.569443\pi
−0.216436 + 0.976297i 0.569443π0.569443\pi
632632 42.4807 24.5263i 1.68979 0.975603i
633633 0 0
634634 0 0
635635 61.2411 + 35.3575i 2.43028 + 1.40312i
636636 0 0
637637 15.9273 + 27.5869i 0.631063 + 1.09303i
638638 0 0
639639 −15.4422 + 28.8898i −0.610885 + 1.14286i
640640 43.3814 1.71480
641641 −8.91478 + 5.14695i −0.352113 + 0.203292i −0.665615 0.746295i 0.731831π-0.731831\pi
0.313503 + 0.949587i 0.398498π0.398498\pi
642642 0 0
643643 7.63202 13.2190i 0.300978 0.521308i −0.675380 0.737470i 0.736020π-0.736020\pi
0.976358 + 0.216161i 0.0693537π0.0693537\pi
644644 40.9582 + 23.6472i 1.61398 + 0.931831i
645645 0 0
646646 0 0
647647 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
648648 −21.1660 + 14.1421i −0.831479 + 0.555556i
649649 0 0
650650 −54.0771 + 31.2214i −2.12108 + 1.22460i
651651 0 0
652652 0 0
653653 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
654654 0 0
655655 −18.5754 32.1735i −0.725799 1.25712i
656656 0 0
657657 0 0
658658 0 0
659659 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
660660 0 0
661661 −25.5444 + 44.2442i −0.993561 + 1.72090i −0.398664 + 0.917097i 0.630526π0.630526\pi
−0.594898 + 0.803801i 0.702807π0.702807\pi
662662 0 0
663663 0 0
664664 25.6919 + 44.4997i 0.997039 + 1.72692i
665665 71.1387i 2.75864i
666666 0 0
667667 0 0
668668 0 0
669669 0 0
670670 0 0
671671 0 0
672672 6.29888 25.1461i 0.242984 0.970030i
673673 25.9422 + 44.9332i 0.999999 + 1.73205i 0.501113 + 0.865382i 0.332924π0.332924\pi
0.498886 + 0.866668i 0.333743π0.333743\pi
674674 37.4166i 1.44123i
675675 10.6442 49.2804i 0.409696 1.89680i
676676 −15.4170 −0.592961
677677 −8.18524 + 4.72575i −0.314584 + 0.181625i −0.648976 0.760809i 0.724802π-0.724802\pi
0.334392 + 0.942434i 0.391469π0.391469\pi
678678 −15.8643 3.97387i −0.609264 0.152615i
679679 0 0
680680 0 0
681681 24.5395 7.00752i 0.940356 0.268529i
682682 0 0
683683 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
684684 −35.7292 + 22.2174i −1.36614 + 0.849503i
685685 −57.3882 −2.19269
686686 22.6826 13.0958i 0.866025 0.500000i
687687 −35.4113 + 36.5892i −1.35103 + 1.39597i
688688 0 0
689689 0 0
690690 60.3226 + 58.3806i 2.29644 + 2.22251i
691691 15.1257 + 26.1985i 0.575409 + 0.996637i 0.995997 + 0.0893857i 0.0284904π0.0284904\pi
−0.420588 + 0.907252i 0.638176π0.638176\pi
692692 49.2050i 1.87049i
693693 0 0
694694 0 0
695695 −7.58667 + 4.38017i −0.287779 + 0.166149i
696696 0 0
697697 0 0
698698 35.0650 + 20.2448i 1.32723 + 0.766277i
699699 −8.48574 + 33.8763i −0.320960 + 1.28132i
700700 25.6710 + 44.4634i 0.970272 + 1.68056i
701701 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
702702 22.4575 24.7774i 0.847604 0.935162i
703703 0 0
704704 0 0
705705 0 0
706706 0 0
707707 −45.9672 26.5392i −1.72877 0.998109i
708708 −17.6397 + 5.03721i −0.662942 + 0.189310i
709709 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
710710 59.2118i 2.22218i
711711 −1.70185 52.0002i −0.0638244 1.95016i
712712 0 0
713713 0 0
714714 0 0
715715 0 0
716716 0 0
717717 27.6791 + 26.7881i 1.03370 + 1.00042i
718718 −9.55452 16.5489i −0.356572 0.617600i
719719 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
720720 21.6907 40.5796i 0.808365 1.51231i
721721 0 0
722722 −36.9527 + 21.3347i −1.37524 + 0.793994i
723723 0 0
724724 3.13802 5.43520i 0.116623 0.201998i
725725 0 0
726726 6.54707 26.1369i 0.242984 0.970030i
727727 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
728728 34.0540i 1.26213i
729729 2.64575 + 26.8701i 0.0979908 + 0.995187i
730730 0 0
731731 0 0
732732 −24.6470 6.17387i −0.910979 0.228193i
733733 25.0267 43.3475i 0.924383 1.60108i 0.131832 0.991272i 0.457914π-0.457914\pi
0.792551 0.609806i 0.208753π-0.208753\pi
734734 0 0
735735 44.7029 12.7654i 1.64889 0.470858i
736736 25.2799 + 43.7861i 0.931831 + 1.61398i
737737 0 0
738738 0 0
739739 0 0 1.00000 00
−1.00000 π\pi
740740 0 0
741741 38.4376 39.7162i 1.41204 1.45901i
742742 0 0
743743 −6.48074 3.74166i −0.237755 0.137268i 0.376389 0.926462i 0.377166π-0.377166\pi
−0.614145 + 0.789193i 0.710499π0.710499\pi
744744 0 0
745745 0 0
746746 0 0
747747 54.4716 1.78273i 1.99301 0.0652268i
748748 0 0
749749 0 0
750750 12.1283 + 42.4718i 0.442862 + 1.55085i
751751 2.77889 4.81318i 0.101403 0.175635i −0.810860 0.585240i 0.801000π-0.801000\pi
0.912263 + 0.409605i 0.134333π0.134333\pi
752752 0 0
753753 2.97356 11.8709i 0.108363 0.432599i
754754 0 0
755755 93.7216i 3.41088i
756756 −20.3725 18.4651i −0.740942 0.671569i
757757 0 0 1.00000 00
−1.00000 π\pi
758758 0 0
759759 0 0
760760 38.0252 65.8616i 1.37932 2.38905i
761761 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
762762 43.4377 12.4041i 1.57358 0.449353i
763763 0 0
764764 4.86804i 0.176119i
765765 0 0
766766 0 0
767767 20.8703 12.0495i 0.753582 0.435081i
768768 19.2728 19.9138i 0.695446 0.718579i
769769 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
770770 0 0
771771 0 0
772772 −20.5772 35.6408i −0.740591 1.28274i
773773 40.8077i 1.46775i 0.679285 + 0.733875i 0.262290π0.262290\pi
−0.679285 + 0.733875i 0.737710π0.737710\pi
774774 0 0
775775 0 0
776776 0 0
777777 0 0
778778 0 0
779779 0 0
780780 −14.6873 + 58.6339i −0.525890 + 2.09943i
781781 0 0
782782 0 0
783783 0 0
784784 28.0000 1.00000
785785 22.2146 12.8256i 0.792871 0.457764i
786786 −23.0213 5.76664i −0.821142 0.205689i
787787 −22.6105 + 39.1626i −0.805978 + 1.39600i 0.109650 + 0.993970i 0.465027π0.465027\pi
−0.915629 + 0.402025i 0.868306π0.868306\pi
788788 0 0
789789 −46.4499 + 13.2643i −1.65366 + 0.472220i
790790 47.0219 + 81.4443i 1.67296 + 2.89766i
791791 17.6648i 0.628087i
792792 0 0
793793 33.3782 1.18529
794794 46.2118 26.6804i 1.64000 0.946852i
795795 0 0
796796 0 0
797797 18.1640 + 10.4870i 0.643403 + 0.371469i 0.785924 0.618323i 0.212188π-0.212188\pi
−0.142521 + 0.989792i 0.545521π0.545521\pi
798798 −32.6555 31.6043i −1.15599 1.11878i
799799 0 0
800800 54.8869i 1.94054i
801801 0 0
802802 45.4422 1.60462
803803 0 0
804804 0 0
805805 −45.3366 + 78.5253i −1.59791 + 2.76765i
806806 0 0
807807 −8.63518 + 34.4729i −0.303973 + 1.21350i
808808 −28.3716 49.1410i −0.998109 1.72877i
809809 31.1127i 1.09386i 0.837177 + 0.546932i 0.184204π0.184204\pi
−0.837177 + 0.546932i 0.815796π0.815796\pi
810810 −27.1134 40.5796i −0.952667 1.42582i
811811 −48.4452 −1.70114 −0.850570 0.525861i 0.823743π-0.823743\pi
−0.850570 + 0.525861i 0.823743π0.823743\pi
812812 0 0
813813 0 0
814814 0 0
815815 0 0
816816 0 0
817817 0 0
818818 0 0
819819 31.8546 + 17.0270i 1.11309 + 0.594972i
820820 0 0
821821 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
822822 −25.4955 + 26.3435i −0.889256 + 0.918836i
823823 −13.2288 + 22.9129i −0.461125 + 0.798693i −0.999017 0.0443211i 0.985888π-0.985888\pi
0.537892 + 0.843014i 0.319221π0.319221\pi
824824 0 0
825825 0 0
826826 −9.90735 17.1600i −0.344721 0.597074i
827827 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
828828 53.5982 1.75415i 1.86266 0.0609609i
829829 1.32653 0.0460723 0.0230361 0.999735i 0.492667π-0.492667\pi
0.0230361 + 0.999735i 0.492667π0.492667\pi
830830 −85.3150 + 49.2566i −2.96133 + 1.70972i
831831 0 0
832832 −18.2026 + 31.5279i −0.631063 + 1.09303i
833833 0 0
834834 −1.35980 + 5.42854i −0.0470861 + 0.187975i
835835 0 0
836836 0 0
837837 0 0
838838 11.6987 0.404125
839839 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
840840 48.2102 + 12.0762i 1.66341 + 0.416670i
841841 14.5000 25.1147i 0.500000 0.866025i
842842 0 0
843843 −55.7336 + 15.9153i −1.91957 + 0.548153i
844844 0 0
845845 29.5576i 1.01681i
846846 0 0
847847 29.1033 1.00000
848848 0 0
849849 −9.22362 + 9.53043i −0.316554 + 0.327084i
850850 0 0
851851 0 0
852852 −27.1806 26.3056i −0.931193 0.901216i
853853 −17.8370 30.8947i −0.610729 1.05781i −0.991118 0.132987i 0.957543π-0.957543\pi
0.380389 0.924827i 0.375790π-0.375790\pi
854854 27.4443i 0.939125i
855855 −42.5953 68.5002i −1.45673 2.34266i
856856 0 0
857857 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
858858 0 0
859859 27.1612 47.0446i 0.926728 1.60514i 0.137969 0.990437i 0.455943π-0.455943\pi
0.788759 0.614703i 0.210724π-0.210724\pi
860860 0 0
861861 0 0
862862 26.4575 + 45.8258i 0.901146 + 1.56083i
863863 14.5365i 0.494830i −0.968910 0.247415i 0.920419π-0.920419\pi
0.968910 0.247415i 0.0795811π-0.0795811\pi
864864 −8.99131 27.9849i −0.305890 0.952067i
865865 −94.3362 −3.20753
866866 0 0
867867 28.5624 + 7.15464i 0.970030 + 0.242984i
868868 0 0
869869 0 0
870870 0 0
871871 0 0
872872 0 0
873873 0 0
874874 88.6348 2.99812
875875 −41.3169 + 23.8543i −1.39677 + 0.806423i
876876 0 0
877877 0 0 0.500000 0.866025i 0.333333π-0.333333\pi
−0.500000 + 0.866025i 0.666667π0.666667\pi
878878 0 0
879879 −24.4014 23.6159i −0.823039 0.796544i
880880 0 0
881881 0 0 1.00000i 0.5π-0.5\pi
1.00000i 0.5π0.5\pi
882882 14.0000 26.1916i 0.471405 0.881917i
883883 0 0 1.00000 00
−1.00000 π\pi
884884 0 0
885885 −9.65737 33.8190i −0.324629 1.13681i
886886 0 0
887887 0 0 −0.866025 0.500000i 0.833333π-0.833333\pi
0.866025 + 0.500000i 0.166667π0.166667\pi
888888 0 0
889889 24.3968 + 42.2564i 0.818241 + 1.41724i
890890 0 0
891891 0 0
892892 0 0
893893 0 0
894894 0 0
895895 0 0
896896 25.9230 + 14.9666i 0.866025 + 0.500000i
897897 −67.7398 + 19.3438i −2.26177 + 0.645871i
898898 −7.78817 13.4895i −0.259895 0.450151i
899899 0 0
900900 51.3420 + 27.4434i 1.71140 + 0.914781i
901901 0 0
902902 0 0
903903 0 0
904904 9.44222 16.3544i 0.314044 0.543940i
905905 10.4204 + 6.01622i 0.346386 + 0.199986i
906906 −43.0220 41.6370i −1.42931 1.38330i
907907 0 0 −0.500000 0.866025i 0.666667π-0.666667\pi
0.500000 + 0.866025i 0.333333π0.333333\pi
908908 29.4685i 0.977946i
909909 −60.1530 + 1.96867i −1.99515 + 0.0652968i
910910 −65.2886 −2.16429
911911 −0.182592 + 0.105420i −0.00604956 + 0.00349271i −0.503022 0.864274i 0.667778π-0.667778\pi
0.496972 + 0.867766i 0.334445π0.334445\pi
912912 −13.3400 46.7150i −0.441730 1.54689i
913913 0 0
914914 48.2404 + 27.8516i 1.59565 + 0.921249i
915915 11.8366 47.2534i 0.391305 1.56215i
916916 −29.3981 50.9189i −0.971339 1.68241i
917917 25.6341i 0.846511i
918918 0 0
919919 −14.4063 −0.475221 −0.237610 0.971361i 0.576364π-0.576364\pi
−0.237610 + 0.971361i 0.576364π0.576364\pi
920920 −83.9471 + 48.4669i −2.76765 + 1.59791i
921921 −58.5038 14.6547i −1.92776 0.482889i
922922 −28.2103 + 48.8617i −0.929057 + 1.60917i
923923 43.0328 + 24.8450i 1.41644 + 0.817783i
924924 0 0
925925 0 0
926926 60.2775i 1.98084i
927927 0 0
928928 0 0
929929 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
930930 0 0
931931 24.5429 42.5096i 0.804361 1.39319i
932932 −34.9230 20.1628i −1.14394 0.660454i
933933 0 0
934934 −29.4365 50.9855i −0.963192 1.66830i
935935 0 0
936936 20.3903 + 32.7910i 0.666479 + 1.07181i
937937 0 0 1.00000 00
−1.00000 π\pi
938938 0 0
939939 0 0
940940 0 0
941941 −35.4825 20.4859i −1.15670 0.667820i −0.206187 0.978513i 0.566106π-0.566106\pi
−0.950510 + 0.310693i 0.899439π0.899439\pi
942942 3.98164 15.8953i 0.129729 0.517897i
943943 0 0
944944 21.1828i 0.689442i
945945 35.4014 39.0584i 1.15161 1.27057i
946946 0 0
947947 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
948948 58.2763 + 14.5977i 1.89273 + 0.474113i
949949 0 0
950950 83.3292 + 48.1101i 2.70356 + 1.56090i
951951 0 0
952952 0 0
953953 29.9333i 0.969633i 0.874616 + 0.484817i 0.161114π0.161114\pi
−0.874616 + 0.484817i 0.838886π0.838886\pi
954954 0 0
955955 9.33303 0.302010
956956 −38.5193 + 22.2391i −1.24580 + 0.719264i
957957 0 0
958958 0 0
959959 −34.2929 19.7990i −1.10737 0.639343i
960960 38.1789 + 36.9499i 1.23222 + 1.19255i
961961 −15.5000 26.8468i −0.500000 0.866025i
962962 0 0
963963 0 0
964964 0 0
965965 68.3308 39.4508i 2.19965 1.26997i
966966 15.9049 + 55.6972i 0.511733 + 1.79203i
967967 −24.2211 + 41.9522i −0.778898 + 1.34909i 0.153679 + 0.988121i 0.450888π0.450888\pi
−0.932577 + 0.360971i 0.882445π0.882445\pi
968968 26.9444 + 15.5563i 0.866025 + 0.500000i
969969 0 0
970970 0 0
971971 31.2787i 1.00378i 0.864931 + 0.501891i 0.167362π0.167362\pi
−0.864931 + 0.501891i 0.832638π0.832638\pi
972972 −30.6732 5.58187i −0.983842 0.179039i
973973 −6.04465 −0.193783
974974 −0.541609 + 0.312698i −0.0173543 + 0.0100195i
975975 −74.1846 18.5826i −2.37581 0.595120i
976976 14.6696 25.4085i 0.469562 0.813306i
977977 51.8459 + 29.9333i 1.65870 + 0.957650i 0.973317 + 0.229465i 0.0736978π0.0736978\pi
0.685381 + 0.728184i 0.259636π0.259636\pi
978978 0 0
979979 0 0
980980 53.6818i 1.71480i
981981 0 0
982982 0 0
983983 0 0 0.866025 0.500000i 0.166667π-0.166667\pi
−0.866025 + 0.500000i 0.833333π0.833333\pi
984984 0 0
985985 0 0
986986 0 0
987987 0 0
988988 31.9104 + 55.2704i 1.01521 + 1.75839i
989989 0 0
990990 0 0
991991 −37.0405 −1.17663 −0.588315 0.808632i 0.700209π-0.700209\pi
−0.588315 + 0.808632i 0.700209π0.700209\pi
992992 0 0
993993 0 0
994994 20.4281 35.3826i 0.647941 1.12227i
995995 0 0
996996 −15.2915 + 61.0460i −0.484530 + 1.93432i
997997 30.9671 + 53.6365i 0.980737 + 1.69869i 0.659533 + 0.751675i 0.270754π0.270754\pi
0.321203 + 0.947010i 0.395913π0.395913\pi
998998 0 0
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 504.2.cc.a.293.1 16
7.6 odd 2 inner 504.2.cc.a.293.4 yes 16
8.5 even 2 inner 504.2.cc.a.293.4 yes 16
9.2 odd 6 inner 504.2.cc.a.461.1 yes 16
56.13 odd 2 CM 504.2.cc.a.293.1 16
63.20 even 6 inner 504.2.cc.a.461.4 yes 16
72.29 odd 6 inner 504.2.cc.a.461.4 yes 16
504.461 even 6 inner 504.2.cc.a.461.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.cc.a.293.1 16 1.1 even 1 trivial
504.2.cc.a.293.1 16 56.13 odd 2 CM
504.2.cc.a.293.4 yes 16 7.6 odd 2 inner
504.2.cc.a.293.4 yes 16 8.5 even 2 inner
504.2.cc.a.461.1 yes 16 9.2 odd 6 inner
504.2.cc.a.461.1 yes 16 504.461 even 6 inner
504.2.cc.a.461.4 yes 16 63.20 even 6 inner
504.2.cc.a.461.4 yes 16 72.29 odd 6 inner